Okay... Got a little farther.
Turns out it's recognized by USB Overdrive as a joystick, not a mouse. It sees both clicks, and tries to move the mouse around by registering it as offsets on the 4 joystick axis's.
When I go into the most recent build of MacMAME, it registers as a mouse. (If I aim and click middle right side, I can get "mouse y" about 60% of the time, and almost always get "mouse x" on middle top side shots. In the game, it just jumps from corner to corner.) But, no clicks. That's with USB Overdrive 10.4 on only. (USB Overdrive calls the clicks joystick buttons 1 and 2.)
When I use 0.66.2 in MacOSX, and I have USB Overdrive 10.4 set to abolute mode, I can get cursor movement which, while not being accurate, _is_ aligned with the gun. But, there are no clicks registered.
The other bad thing is that the gun trigger is on a slide, and not a pivot pin. Stupid design. Just from the few hundred test fires, it's already kind of "dragging" as I pull unless I hold it just right. I might likely return it because of that, whether I get it working in MacMAME or not.