AFAIK, the golf cheat is all internal to the USB board, and just forces zeros on the X axis before it's sent over the USB cable. As said, no software at all.
The middle button is a funny thing in windows. Sometimes it needs a special driver, sometimes the three button works with the driver windows automatically gives it. (Remember, "no drivers" means "windows already has a driver that works with this device".) 99% of windows apps don't use the middle button, and most drivers can, and do, change what pressing the middle button sends applications. Mame is one of the few apps that read mice through RawInput, which is "lower" than most driver middle button remaps, so the question is, in mame, can you map the middle button? (What about the other mouse's middle button?) Unless you know the difference between a middle button press in windows and the action a remapped middle button, don't try to compare the TB and the mouse's actions in windows.
If you can't remap middle button in mame, and the golf cheat doesn't work, sounds like either a wiring error, or the TB's USB board isn't working like it should.