So I found a bug in MAME. I'm using a GPWiz40, and if I set an input to Button 16 in MAME, quit the game, and then restart it, the input will be set to Button 15. So basically, I have to reset the input every time I start the game to Button 16.
I attempted to issue a bug report to the MAME team:
http://www.mametesters.org/view.php?id=3392. The issue appears to be stuck on "Direction needed." I'm not quite sure what this means. Are they waiting on more direction from me?