Maybe Henry would want that, but Scotland would NOT. Meaning who an Henry with three sons can easily offer to Scotland the choice among his sons, specially if he has already planned another very prestigious match for his heir (who is much older than the Scottish Queen)…
If Henry VIII had more boys Scotland can easily reject the Prince of Wales for their infant Queen but offer a match with one of the younger brothers, likely the youngest, who can be raised at least partially in Scotland together with his bride-to-be. A second or better third son of the King of England would be, in my opinion, a much more attractive match for the Scottish than either the Prince of Wales or the Dauphin as would consent to them to keep their independence
But why would Henry VIII do that?
In OTL, he tried to
kidnap Mary in order to force her to marry Edward. Why wouldn't he do the same thing here?
Also, there is something very important you do not seem to take into account: Mary's claim to the throne of England. In the eyes of the Catholic Church, any child of Henry VIII and Anne Boleyn is a bastard and Mary Stuart becomes the legitimate queen regnant of England at the second Mary Tudor dies (except of course if Mary Tudor has a legitimate child). Considering that, how could Henry VIII propose a younger son for Mary Stuart? That would be the best way to end up with a new War of the Roses between two Tudor branches.
My guess is that Mary Stuart's fate, at least at the beginning, would follow the OTL path.
Henry VIII tries to force a marriage with his eldest son. To prevent that, Marie de Guise sends Mary to France. Mary marries Francis who still dies young. Then, depending on Henry II's death being butterflied away or not, Mary may either marry Charles (which would be very interesting) either come back to Scotland. If she comes back to Scotland, I guess she can marry William but only if William rebels against Henry IX, converts to Catholicism and claims the throne of England in the name of his wife.
If Henry VIII is reasonable and does not try to force a marriage (but it sounds very out of character), I guess the Scottish would keep the Auld Alliance, but not to the point of a personal union with France. Then, Mary is likely to marry a French prince but not the dauphin. François de Montpensier would be perfect.
Well I think who at that point England was too firmly in the Imperial/Spanish camp for having Henry II interested in marrying his son to an English princess, specially when the youngest is four years older than hi . That or maybe Dauphin Francis died early, leaving a Dauphin way too young for her
Francis always had terrible health and he is likely to die early anyway. I think he would still marry Mary but, if he does not, I don't think he would have such an obvious fiancee and I guess he would die before marrying (in OTL, he died at the age of sixteen).