By which I mean, is there a way to better render a model so that nearby polygons aren't sorted incorrectly and don't appear to override each other? In this particular case, I'm having trouble with a cape which animates fine in Max but which appears to move into the character in jpct.
Sounds like depth fighting to me. That depends on the accuracy of the graphics chip's zbuffer. You can try to lower the far clipping plane if that's an option, but usually, this doesn't help much.
So the only solution is to change the animation itself?
Yes, basically. As said, you might get some improvement if you adjust the far plane, but to be honest, it never helped for me albeit in theory, it might.