Maybe I'm misunderstanding what's going on here, but can't you setup the the U360 as the default joystick for all machines in the MAME debug menu? That way every game you have would use that as your default Player 1 and then you could change individual machines if you wanted after that.
Yeah, I totally forgot that MAME will take multiple inputs for a specific function.
So all I had to do was Tab into the settings of one game, pick Input (General) and add the U360 to the player 1 directional values. Now most games are set...
The only one giving me a tiny headache is Food Fight. Took some research to realize a few things. 1) The AD Stick "Inc" and "Dec" values in the Player 1 Controls settings had to have Joy 1 deleted. 2) I had to dial back both Joystick_deadzone and Joystick_saturation in the general Controllers menu of MAME32UI.
The only issue is that Food Fight is still not perfect. The menu still wants to jump back to the left automatically when you start a game. I tweaked both of those values a bunch and it either stays like that, or wants to automatically go right. It won't stay put. It's a small issue because the main thing is the game, and the character responds just fine.
Sinistar oddly works perfectly.