Proposing my own plan, cutting out the ambiguities.
[X] Plan Stark Reality
-[X] Tell Stark that you would prefer to let him grieve in peace, but there are matters that need addressing urgently and thus it can't be helped. Jon should be present for this since most of it concerns him.
-[X] In regards to Cersei and the Iron Throne:
--[X] He has gotten the news from the south and if he does not believe you or your newspaper, then you are perfectly willing to perform the divinations confirming the relevant information right here and now in his presence. Or wait for his mages to do the same.
--[X] His feelings for his friend Robert are one thing, but the facts are that the children are not his, the woman who murdered him is now sitting on the throne and it's anyones guess what other atrocities she will unleash in the coming months. For all you know, they will try to bind further Winter Fey, or mayhaps outright pact with them, opening the door for more and more such assassins to slip through the gaps int he First Mens defenses.
--[X] He will have to consider where he stands since this has implications...
-[X] In regards to Winterfell and his allegiance:
--[X] So far, you considered the wards of the castle sufficient to protect the inhabitants, but apparently that is not the case. This has both personal implications in regards to Jon and strategic implications, as Winterfell may or may not hold a key to a vulnerability of the Wall itself. It's defenses must be increased.
--[X] You will not pretend that any troops you send would be wholly under Starks control and so that fig-leaf will not even be considered. If he accepts these forces, then that means he swears fealty to the Imperium and accepts the terms offered in regards to the political restructuring and dissolution of the North as recompense for damages incurred due to his participation in the Rebellion.
--[X] If he accepts, you will send the City Defense Group over immediately, including some selected Inquisition operatives who will begin to establish an Inquisition presence in Wintertown, Winterfell itself and the countryside. You will send further forces once you had time to review current deployment orders.
--[X] If he refuses and wishes to stand with the Lannisters, then you will henceforth have to treat him as an enemy. You were willing to let him sit on the fence for the longest while due to what he has done for Jon, but there is a time when you can no longer afford such indulgence. Though you will swear oath before the Heart Tree of Winterfell that no harm will be done his children by you or anyone loyal to you if he chooses to be your enemy, neither Sansa nor any who remain in Winterfell and neither will they be used as bargaining chips in one way or another. You are not Tywin Lannister.
--[X] However, he will have his time to grieve before deciding. You expect his answer in one week at the latest. If he does not answer, then you know where he stands and wish him the best of luck on that path.
-[X] In regards to Lyanna:
--[X] Confirm that you are aware of her ghost having existed in Jons head for the last few years. This should be uncontroversial, since Stark saw her manifestations himself.
--[X] You have not brought up the matter before, due to the political ramifications of revealing Jons parentage, but those considerations a moot now.
--[X] You have no feelings one way or another on Lyanna Stark, but Jon deserves to have privacy in his life, not the constant presence of an undead spectre. Therefore, you propose the following options:
---[X] Returning Lyanna to life with True Ressurection.
---[X] Transferring her soul to an artificial Warforged body.
---[X] Performing arcane rituals upon her to stabilize her soul and prevent her state of being from affecting her mental faculties while disentangling her from Jon. (Turning her into a free, Soulforged undead.)
---[X] Putting her to final rest, sending her soul on to the Green Dream.
--[X] The choice which path she wants to take is Lyannas, but it will have to be one of those options, though Stark can of course offer his input on the matter. The current state of affairs is unhealthy for Jon, suboptimal for Lyanna, and both of them would likely be much happier with another arrangement.
-[X] In regards to Jon:
--[X] You hereby offer him to become your ward, which includes him getting a place to study at the Scholarium. Beyond that, this will include combat training, advanced studies and the necessary equipment for his safety and desired path in life. In the time that Stark wants to take to consider his options, you would take him to SD so that he can see what awaits him there. You will also throw in a bag of candy, because you are perfectly aware of what you are doing, but the ends justify the means.
--[X] That choice is Jons alone.
---[X] If Lyanna wishes to accompany him, she can do so and will be hosted as a guest of the state.