FBA needs specific ROMs for it, and each of the MAME cores requires its own matching ROMs, as well. That’s just the way it goes with arcade ROMs, unfortunately.
If you load up the core(s), the exact version should appear next to the core name in the bottom-left of the screen. Look for ROMs from that specific set and they should work fine.
None of the main team members use emulationstation so we can’t provide any guidance on that but maybe someone else can chime in.
Btw, I edited your post to remove the link to the romsite, which isn’t allowed.