Just one point - Richard de la Pole was recognised as head of the House of York until he died at Pavia in 1525, as his older brother had been designated heir to the throne by Richard III.
I know that your Withered Rose timeline has him as a prime protagonist, so you're partial to him. But I have to wonder how partial the English nobility would be. For the sake of argument, let's assume that your invasion isn't the POD here, just Henry's death.
As far as I can see it, with the exception of Buckingham, de la Pole has the weakest of all the Plantagent-derived claims, as his claim descends from a sister of Edward IV, not one of his daughters, and dates back to the designation of Richard of not-so-beloved memory in 1484. Even then his claim depends on the acceptance that all of Edward IV's children were illegitimate. I don't think that was the general case in the England of the 1510s. By then, too many people had a vested interest in asserting the legitmacy of Edward's children: Henry VIII, Margaret Tudor, James Stewart, Mary Tudor, Louis XII, Suffolk, Exeter, just to name a few.
To me, de la Pole's claim to be the head of the House of York seems very similar to the claims of latter day Jacobites; more than a little divorced from the realities of the England he had left a decade before. He had a claim, it was useful for some foreign powers to use him as a tool from time to time, but I have to wonder how that very support would play out in England. His claim would be a fig leaf that would try very hard to hide the fact that he needed a FRENCH army to put him on the throne. Granted it worked for Henry VII, but that was a slightly different stuation, in which Richard III was rapidly wearing out his welcome.
This isn't the case here. There was no national mourning or rebellion when Henry VIII came to the throne. Even the Yorkists accepted him as the union of the two houses. Most of his problem with them came from his own paranoia. His death, presumably in France, would occasion an outpouring of grief and a search for stabiity and legitimacy that would avoid anything that smacked of a civil war.
James I/V would be the living symbol of that unity, and its extension to peace with Scotland after the events of Flodden. It's even a win-win. The Scots get the title, the English get the power, the nobles get a bit of freedom, Ireland gets left alone. In fact, I'm not sure if Francis I (assuming its him and not Louis XII) wouldn't just breathe a sigh of relief, count his blessings, ferry the English back to England, re-claim Calais at the treaty table, pension off de la Pole, consider a marriage proposal between a future daughter and James I/V for down the line and then turn his attention back to Charles V. A French backed invasion by de la Pole would cost money, divert resources, and re-open a war on that front. It would also be just what the new regime would need to solidify itself and its legitmacy (and emnity to France).
Even if the Yorkists decided for some reason to try to turn back the clock and ignore anything Tudor, de la Pole wasn't the legitimate claimant by the 1510s. If I'm not missing a sister somewhere, Henry Courtenay had the senior claim after The Tudor Girls, thorough his mother Catherine of York, and he had the friendship of Henry VIII, so he'd pull some Tudor loyalists to his side. Given his connection to Norfolk, I figure he could swing the support of the other nobles and possibly the throne.
Now that's an interesting option for your timeline...a three way civil war between Courtenay/Norfolk, the Tudor-Stewart claim and your French-backed de la Pole (maybe with a marriage to one of the Tudor Girls to bolster things).
David