Question: If we are not to petition during the Prayer of the Heart (PH) and Lectio Divina (LD), when and how should intercessory prayer happen/take place?

Answer: As an allocated moment (time and space), intercessory prayer should happen at a special time, as a specific type of prayer. During Mass we have a time for it, as we do during the Divine Office (I am sure that in our personal prayer as well we do pray for others). Your may ask the same question for other types of prayer as well: Prayer of Praise and Prayer of Thanksgiving, “when shall we say them?”.
Here I would like to explain more clearly the deep relationship between LD & PH on the one hand and Intercessory Prayer on the other.

It is important to remember that LD is a key unavoidable type of prayer today. It is by far the most transformative type of prayer. Why? Because it allows a part of Christ (today’s part of Christ-Word Bread) to become incarnated in us and transform us in Him. The more we are transformed in Christ the more powerfulour intercession becomes (acts on God).

Jesus explains the relationship between Lectio Divina and the efficiency of our prayer (which has been answered) in the Gospel of St John where He says: if you do my Will (keep my commandments) ask me all what you want (because you’ll BE in my Name) and you will receive it, i.e. the Father will grant it to you (see John 15:7). If LD is done properly, all its transformative power will be enacted. Then PH becomes much more fruitful and even more transformative. Otherwise, if you practise PH without LD, its transformative power isdramatically decreased.
Now let us consider what happens during the PH. While we are practising the PH, the Power of the Holy Spirit is working in our depths and through us will help Lift the entire World to God the Father in the Son, through the Holy Spirit. This is done automatically. The more we are transformed in Jesus, the better it works – automatically. St Therese explains that wonderfully at the end of her Manuscript C, in the Story of the Soul.

She says:

“All the saints have understood this, and more especially those who filled the world with the light of the Gospel teachings. Was it not in prayer that St. Paul, St. Augustine, St. John of the Cross, St. Thomas Aquinas, St. Francis, St. Dominic, and so many other famous Friends of God have drawn out this divine science which delights the greatest geniuses? A scholar has said: “Give me a lever and a fulcrum and I will lift the world.” What Archimedes was not able to obtain, for his request was not directed by God and was only made from a material viewpoint, the saints have obtained in all its fullness. The Almighty has given them as fulcrum: HIMSELF ALONE; as lever: PRAYER which burns with a fire of love. And it is in this way that they have lifted the world; it is in this way that the saints still militant lift it, and that, until the end of time, the saints to come will lift it.”

Intercessory prayer has different levels of power and action. What is more powerful than to pray for somebody or – if it was given to you by the grace of God – to take this person and lift him/her to God and introduce him/her into God’s Fire of Love? Here St Therese presents to us the most powerful version of Intercessory Prayer.It is a duty of love for us to pray for everybody. St Paul says that we have to pray for one another all the time, without excluding any other person. St James in the end of his Letter, mentioning the Prophet Elijah speaks about the power of a prayer that is heard/answered by God. This should grab our attention and invite us to deepen our understanding of Intercessory Prayer.
In addition, one has to remember the relationship between Intercessory Prayer and the  Priesthood of the Faithful. A Priest “prays for” others, like St Paul invites us to do all the time. This is why we have the Divine Office. But it is something that becomes second nature to us, it is part of the “fabric” of the New Creature that we become in Christ. Being “in Christ” makes the Fire of the Holy Spirit that dwells in Him, pray in us and through us. The Holy Spirit knows how to pray and knows what we should ask for, and the way God wants it. What depends on us is not to pay attention to all that multiplicity of things to pray for. Our duty is just to get closer and closer to Jesus and to the Fire of His Love and be transformed by IT.

St Therese developed those important questions in two different places in that same Manuscript C, in the Story of the Soul:

“Since I have two brothers and my little Sisters, the novices, if I wanted to ask for each soul what each one needed and go into detail about it, the days would not be long enough and I fear I would forget something important. For simple souls there must be no complicated ways; as I am of their number, one morning during my thanksgiving, Jesus gave me a simple means of accomplishing my mission.He made me [34r°] understand these words of the Canticle of Canticles: “DRAW ME, WE SHALL RUN after you in the odour of your ointments.” O Jesus, it is not even necessary to say: “When drawing me, draw the souls whom I love!” This simple statement: “Draw me” suffices; I understand, Lord, that when a soul allows herself to be captivated by the odour of your ointments, she cannot run alone, all the souls whom she loves follow in her train; this is done without constraint, without effort, it is a natural consequence of her attraction for You. Just as a torrent, throwing itself with impetuosity into the ocean, drags after it everything it encounters in its passage, in the same way, O Jesus, the soul who plunges into the shoreless ocean of Your Love, draws with her all the treasures she possesses. Lord, You know it, I have no other treasures than the souls it has pleased You to unite to mine; it is You who entrusted these treasures to me, and so I dare to borrow the words You addressed to the heavenly Father, the last night which saw You on our earth as a traveler and a mortal. Jesus, I do not know when my exile will be ended; more than one night will still see me singing Your Mercies in this exile, but for me will finally come the last night, and then I want to be able to say to You, O my God: “I have glorified you on earth; I have finished the work you gave me to do. And now do you, Father, glorify me with yourself, with the glory I had with you before the world existed.“ I have manifested your name to those whom you have given me out of the world. They were yours, and you have given them to me, and they have kept your word. Now they have learned that whatever you have given me is from you; because the words you have given me, I have given to them. And they have received them, and have known of a truth that I came from you, and they have believed that you sent me.“I pray for them, not for the world do I pray, but for those whom you have given me, because they are yours; and all things that are mine are yours, and yours are mine; and I am glorified in them. And I am no longer in the world, and I am coming to you. Holy Father, keep in your name those whom you have given to me.
“But now I am coming to you; and these things I speak in the world, in order that they may have joy made full in themselves. I have given them your word; and the world has hated them, because they are not of the world, even as I am not of the world. I do not pray that you take them out of the world, but that you keep them from evil. They are not of the world, even as I am not of the world.
“Yet not for these only do I pray, but for those who through their word are to believe in me.
“Father, I will that where I am, these also whom you have given me may be with me, that they may see my glory which you have given me, because you loved me from the foundation of the world. And I have made known your name to them, and will make it known, that the love with which you loved me may be in them, and I in them.”” (Manuscript C)

In fact Thérèse is explaining the Common Priesthood of the Faithful, received in Baptism, in Jesus-Priest. And a few pages afterwards she gives a further explanation of her new way of praying, her new way of practising the Prayer of the Heart:

“Mother, I think it is necessary to give a few more explanations on the passage in the Canticle of Canticles: “Draw me, we shall run,” for what I wanted to say appears to me little understood. “No man can come after me, unless the FATHER who sent me draw him,” Jesus has said. Again, through beautiful parables, and often even without using this means so well known to the people, He teaches us that it is enough to knock and it will be opened, to seek in order to find, and to hold out one’s hand humbly to receive what is asked for. He also says that everything we ask the Father in His name, He will grant it. No doubt, it is because of this teaching that the Holy Spirit, before Jesus’ birth, dictated this prophetic prayer: “Draw me, we shall run.”What is it then to ask to be “Drawn” if not to be united in an intimate way to the object which captivates our heart? If fire and iron had the use of reason, and if the latter said to the other: “Draw me,” would it not prove that it desires to be identified with the fire in such a way that the fire penetrate [36r°] and drink it up with its burning substance and seem to become one with it? Dear Mother, this is my prayer. I ask Jesus to draw me into the flames of His love, to unite me so closely to Him that He live and act in me. I feel that the more the fire of love burns within my heart, the more I shall say: “Draw me,” the more also the souls who will approach me (poor little piece of iron, useless if I withdraw from the divine furnace), the more these souls will run swiftly in the odor of the ointments of their Beloved, for a soul that is burning with love cannot remain inactive. No doubt, she will remain at Jesus’ feet as did Mary Magdalene, and she will listen to His sweet and burning words. Appearing to do nothing, she will give much more than Martha who torments herself with many things and wants her sister to imitate her. ” (Manuscript C)

All that Thérèse is describing happens in the same movement of the Prayer of the Heart. Jesus attaches people to us (without us doing it or knowing it), so when we do the PH we perform, as well, our Priestly duty of intercession. In the Prayer of the Heart we offer our being to Jesus and are immersed in Him. The more we are transformed in Jesus the more we are like a sponge, unknowingly absorbing all the persons Jesus wants us to carry. They are like our mystical body (“Union with Jesus” is union with a portion of his Body). So whenever we do the Prayer of the Heart (The more we are transformed in Jesus, the more PH becomes more and more constant in us), and we are then immersed (with them) in His Fire. The Power of the Holy Spirit in us is lifting not only us but all our mystical body.[Many more things occur during the PH, and this is just to reply to your question and show the deeper levels of Intercession and their relationship with our level/degree of transformation in Jesus and with the practice of the PH.]In conclusion we can say that it is not really possible to separate deep powerful intercession from our being, from our PH.

When somebody asks us to pray for him/her, or we just remember to pray for a certain person/intention, let us not forget that (in my eyes) the most powerful, perfect and pure way to do it is to entrust this intention to Our Lady, by saying one “Hail Mary” and that’s it. If your remember again this person/intention, you may just redo that again: say the “Hail Mary”, entrusting this person to Mary. St Therese used to say that Our Lady knows better the will of God (what to say to God and how to say it), we don’t. In this sense she is called “Throne of Wisdom”, because the Wisdom of God, Jesus, is dwelling in her in His absolute perfection while it is not the case for us. We either carry too much the person, or worry too much, or are busy too much in dealing and arranging the life of others, forgetting our utter ignorance. And all this is not intercessory prayer, but impurity added to our way of dealing with God.
I hope this helps.