What Scotland ABSOLUTELY do NOT need is another costly military defeat against England and Burgundy so if they are not total idiots will NOT continue to stay in the French camp as that would make them an easy target
This is 100% a case where bandwagoning is the better way to go. There is no way the Auld Alliance can keep Scotland independent anymore. England and the Low Countries combined would need only a fraction of their wealth to raise an army large enough to crush Scotland, and they would still have plenty left over to raise an even bigger one to fend off a French attack on the Low Countries. Not to mention this Anglo-Dutch union is a part of a Habsburg triumvirate encircling France from all sides. If France attacks the Low Countries Spain will attack France in turn.
And yet, the Habsburgs and the Tudors did not "crush" Scotland in OTL. Neither during the Italian War of 1542–1546 (that coincided with the Rough Wooing and in which Henry VIII and Charles V were allied against France), neither when Mary Tudor and Philip II were married (while Mary Stuart was in France and engaged to the future Francis II).
If the Habsburg/England alliance could so easily "crush" Scotland, why didn't it do it? The Habsburg/England alliance DID exist in OTL. Despite this, France and Scotland survived and the Auld Alliance lasted until the Treaty of Edinburgh in 1560.
Why would it be different ITTL?
Also France really wasn't in any position to invade anybody to begin with. The French Wars of Religion kept them busy for the remainder of the century, and really it wasn't until the outbreak of the 30YW that France started launching large invasions into other places again.
Sure. With the Treaty of Cateau-Cambrésis, Henri II let to the Habsburgs all the territories they disputed (save from the duchy of Burgundy). After that, the Valois seemed to have renounced to the idea of openly challenging the Habsburg's hegemony and they tried to expand their influence through more indirect ways: election of the duke of Anjou as king of Poland, election of the duke of Alençon as "protector" of the Netherlands WITHOUT his brother's official support, marriage proposals to Elizabeth I, support to the prior of Crato for the Portuguese throne...
Likely, things would turn like that ITTL too.
As for the Scottish, they may or may not end the war before the French and they may or may not concede a marriage to a duke of York but NOT to a prince of Wales. And that would not be the end of the Auld Alliance even if open war is over.
In case of the whole Scotland thing, I can’t help asking whether a marriage alliance with Denmark-Norway would not be better for them, there’s of course the religious aspect working against it. But Denmark-Norway is a rather strong state at this point and not one in conflict with Habsburg or England, so both the English and Habsburg would likely be careful about pushing it into alliance with France.
Ah, the Denmark-Norway thing...
Charles V's nieces Dorothea and Christina were claimers to its throne. Christina actively tried to push her claim as late as in the 1560s. King Eric XIV of Sweden accepted to help her if she managed to also get the support of the emperor and the Netherlands. The emperor refused because that could endanger the equilibrium of powers in the Holy Roman Empire, due to Saxony being allied to Denmark-Norway.
Could Mary Tudor (if she is still alive ITTL), being the religious zealot that we know, push the claim of her Catholic cousin?
Interestingly, Christina's son and heir Charles of Lorraine had been raised in French court, was married to a French princess and was good friend with his brothers-in-law, making his mother's claim rather ambiguous towards the Habsburgs. Of course, this point can easily be changed ITTL.