Obviously Edward is a medieval monarch no some proto-Ban Ki Moon. As I said earlier is number one priority is going to be serving his own interests whether that's expanding his domains or saving his soul by killing infidels. You can see that in Wales where a Civil War, Welsh involvement in the Baronial conflict and cross-border raids focused his attention on the area and his pride and ambition demanded he did whatever it took to win.
Forget Ban Ki Moon (don't get the reference, I'm afraid). This is Edward the Conqueror, with all of what that means.
I've got to disagree. He was happy with that from Alexander III and while he might like more he can live with maintaining the status quo.
Even homage for Alexander's English territories alone is more than just "nominal". And unlike the hypothetical Alexander IV, we're dealing with an of age and reasonably secure (internally) king, which is not offering an easy opportunity the way OTL or TTL is. So I can see Longshanks having more interest in squeezing the grandson than the grandfather.
To an extent I agree with you. If he gets involved his pride and his ambition means he will follow an OTL course of demanding that whoever ends up on the Scots throne is an actual rather than just theoretical vassal. But I don't see how that makes he bound to intervene in Scotland. Think about the OTL course of events. In 1286 Alexander III dies and leaves the Maid of Norway as heir. Edward does nothing about it and around that the time that he leaves for France to try and resolve the extremely complicated situation in Gascony He stays for three years trying to hold things together and only papers over the cracks. The situation in Scotland with an absentee Queen and feuding Regents leads to returning in 1289 when he oversees the
Treaty of Birgham where Edward succesfully arranges the marraige of the Maid of Norway to Edward II. Then she dies and the whole thing blows up.
And if Alexander III's heir is instead an underaged son (who does live, unlike the Maid), I don't see Longshanks using that to squeeze as hard as he can the way he squeezed John OTL. Not immediately - as you said he's dealing with Gascony, which is several times more important - but if this hypothetical son is born in 1284, and is thus two when his grandfather dies - there's plenty of time for Edward exploiting an underaged king for all he can get while dealing with the immediate Gascony situation.
Now put yourself in Edward's boots in 1289. You're in Gascony, while you have made major strides in stabilising the administration the new French King Philip IV is looking for any excuse to confiscate your Duchy. He's extremely competent and is busy turning France into a centralised efficient Kingdom like England. You are aware that Scotland has a boy King but there is no real prospect of a Royal Marriage bringing Scotland into your realm (though you may still try and negotiate a marriage between Edward II and Margaret, its an alliance that makes sense). You've also got your Crusade project, it's on the backburner at the moment but you fervently believe that the path to heaven is paved with infidel bodies.
Now do you go home to Britain and start causing trouble with Scotland or do you make a flying visit to Westminster to extract some more money from Parliament and remind your Barons that you're still number one and then head back to France whether to fight Phillip or organise a Crusade?
The problem is that it isn't as if Edward can't do #2 now and #1 later when the situation with Gascony isn't demanding his immediate attention. Edward was in fairly good health most of his life, which means that (going with Alexander IV being born in 1284) he has over a decade between 1289 Alexander IV being of age and at least slightly harder to push around.
So if I was in Edward's shoes and acting as Edward did: Deal with Gascony and Philip now (making a quick trip to England to knock heads together and say "I am the king. Gimme my subsidies." if needed), see if the crusade idea can be revived, and take advantage of any opportunities to exploit Alexander IV's position for all they're worth when they come up.
I don't have to pick "deal with Gascony and forget Scotland" or vice-versa. And in Longshanks's position, I doubt he would.