To be honest the more I think about how Bahamut and the egi operate the more I realize their underlying AI hasn't changed at all. This is mostly from trying to pry out the development process that went into him. Namely, how little would SE have to change to make him work. The Egi while on Heel will immediately return to you once your target is dead or gone and they haven't been given a new one, but they will remain still and auto attack as long as they have a target, even one presented to them within their GCD timer. Movement commands also interrupt Egi actions and can even eat cooldowns like Devotion if you don't think about it.

The only thing Bahamut does differently is he doesn't auto attack. Thus a lot of his behaviour makes sense if they went for a very minimal change (just removing the auto attack portion so they could add the wyrmwave proc) and left him in the default 'Heel' AI. I also know he doesn't inherit your pet's AI stances, as a place command on an Egi is enough to override that behaviour but any resummon immediately defaults back to Heel, so there's no way to override that behavior. I also know from playing with him before that he remembers which Egi you last summoned too, even one you attempt to summon while he's out, but that can't be used to do anything since he overrides the egi until his timer expires.

The overall Egi and Bahamut AI thus needs to be changed to default to standing still as if affected by a Place command, unless commanded to attack an enemy out of line of sight, instead of moving next to the Summoner immediately once a target is lost, with Heel being treated as a movement command that pulls the pet to a random position within the Summoner's range. That would fix all of their problems.