I am noticing that MAME does not like combo buttons... I have start plus player 1 button 1 set for coin and player 1 start as start for player 1. MAME does not like this... It enters credits when player 1 start is held and tapping player 1 button 1, but as soon as you release the start button it accepts it as start. So I cannot start 2 player games with it mapped as suggested above. 
Have you tried holding down P1B1 and then tapping P1 start with this setup? Mame doesn't care which button is pressed first, just if both are down at the same time, and P1B1 usually doesn't do anything until a game starts playing. Hmm... you still might dave problems unless you map start as "P1 Start Not P1B1" also. (To do a Not, press the button you don't want twice quickly after you pressed the button you do want: enter, P1 Start button, P1B1, P1B1.)
Even with this "Not" part added, however, if you press the button order like you discribed, you'd still have the two player problem. And the "Not" probably won't help your shifted u360 config. I say probably because it
might work if: you press and hold start, press and hold P1B1, release start, then release P1B1. I say might because if depends on how fast the u360 updates it's state, how fast/often mame polls the state, and maybe even when you release the start button compared to the prior two.
Also, this means you have to be careful what combos you map,
especially for exit (default mapped to esc key).