Would it be bad for an Axiom sub to purchase a stronger amplifier for it? Would it have any real performance benefits?

No. A sub, or any other speaker, uses only as much power as it needs at the moment, regardless of how much more is available. Unless the Axiom sub amps aren't adequately powerful, which isn't the case, there's no reason to think of such a project.

I believe you would be virtually changing the sub design. I would not feel comfortable replacing the amp without consulting Axiom. It may not mean anything as JohnK said but it could also pose a potential problem.