I’m convinced it’s a pride thing, the Greens stood against them in the 2017/19 and they’re holding onto that grudge.
This will be an unpopular comment here, but narcissism really is Corbyn's dominant personality trait, so you may well be right.
But even so, there is an old school Marxist way of looking at political parties, which is that any member-funded party is de facto materially a bourgeois party regardless of what its policy platform is. I think a lot of people on the left share that view, which has always limited the appeal of the Greens to Labour's left flank. I think that if Corbyn had his own way, he'd set up a party along the lines of Labour's original organising principle, in which there was no individual membership; you could only join if you were a member of an affiliated union or a socialist society (like the Fabians, which basically became a way for middle class people to join the Labour party!).
Again, though, this is a case where both explanations - pride/narcissism AND genuinely held philosophical views - can be true!
They really should just formalise Momentum as a party, tell the Momentum-affiliated Labour MPs to jump or be pushed, and use Momentum as the name. Already has recognition, for one thing.