To clarify for anyone confused, the 8 button inputs show up in windows as joystick buttons, not keys. So if you ONLY use the u360 for your CP, you can use up to 8 buttons, run them into the U360, then plug the U360 into the computer, which will then show up in windows as a joystick and 8 joystick buttons (or rather 15 joystick buttons if you count the shift and 1-7).
The problem here is that if you unplug your U360 and plug it back in, Windows might assign it to Joy1, Joy2, or whichever port it decides, and now any settings in mame might be wrong. One way to get around this is if you are using a separate encoder and use those inputs as joystick "switch" outputs. This makes it look (to the encoder) like any 4/8 way joystick. The result is keypress outputs for your up, down, left, and right. You can still program the u360 map for easy diagonals, 4-way only, etc, but the output is just simple digital up, down, left, right outputs, and regardless of what joystick number the PC assigns, you will never have to reprogram mame if you mistakenly unplug that stick. However, then you need an encoder with enough inputs for both joystick and buttons.