Any finitely presented simple group has solvable word problem, and hence recursive Dehn function. I'm curious though how wild these recursive functions could possibly be.
One concrete question is whether there are even any examples known of a finitely presented simple group whose Dehn function is exponential. The only examples I can think of at the moment of finitely presented simple groups where something is known about their Dehn functions are Burger-Mozes groups (quadratic Dehn function) and Thompson's groups $T$ and $V$ (polynomial Dehn function, conjecturally quadratic).
Another (vaguer) question is whether there is some stronger bound on what sorts of functions can be Dehn functions of finitely presented simple groups (stronger than just being recursive, I mean). E.g., maybe there's some reason they can't be super-exponential?