I don't use MaLa or Fatfinger's pluggin, so take this with a grain of salt. Also I haven't seen your prior post, so I don't know what's been suggested. Anyway...
Try setting the u360 in analog mode for all games
all the time, and let mame do the automatic mapping. Mame can do exactly the same thing as u360 is doing, without need for a pluggin. If you didn't change u360's default maps (8way/4way/4way diagonal), you shouldn't notice a difference.
If you did change the maps, you can give mame the equivalent; for now just test with mame's defaults, and I'll help you with mame's analog to digital mapping format later.
I am assuming you are using a recent or current mame, of course.

My guess at what's going on with the little info I have: the u360 is restarting after the remap, and mame starts before the u360 recovers. Since mame only looks for new joysticks at startup, it misses joy1. (I wonder if you can add a delay between remapping u360 and starting mame?)
Some questions I have if I'm wrong:
Player1 u360 is joystick 1, and p2 is joy2?
Sometimes joy1 works, sometimes it don't, and a game might flipflop between joy working and not? Or once a game doesn't work once, it continues not working?
Does joy2 act is if it was joy1 in the joy1 not working times? Or is it still joy2?
Something else you also might want to do:
In mame "input (general)" "user Interface", map both joy1 & joy2 as UI_left & UI_right. Then you can wiggle either joy1 or joy2 to get past the ok screen.