I'm not saying he can't. And I'm sure he'd be interested since he used the Superman reference in The Iron Giant, but based on that and the tone of The Incredibles, and the fact that he likes to work from his own scripts, there's a lot of heavy lifting between where Clark was in MoS and BvS that isn't simply a matter of being 'serious' or 'dramatic' or whatever you want to consider the opposite of 'fun.' Bird's films aren't known simply as 'fun' films - there were plenty of serious and heavy messages in all the Pixar movies he made and in Tomorrowland, which is a better example than Mission Impossible 4.
The only thing MI4 suggests is that he would be willing to work within a world-view that is not his own, inside an existing and previously built universe, with previously set casting and other constraints.
All that said, my point stands: Bird can turn things down, he accepted MI4 because he wanted to prove he could do live action. Tomorrowland didn't change that, it just showed that his name, Clooney's name and interesting looking science fiction wasn't going to be enough to make Interstellar or Inception dollars and he's not going to get that kinda of a budget on a passion project or something original that isn't Pixar.
He's also busy on The Incredibles which has a release date and another movie for Pixar, which doesn't, so he's not a realistic possibility for anything in the next 3 years at minimum, probably longer.