At those weight differences you should almost be able to tell which figure is correct by comparing fuel flow with a similar A320 at the same weight
I'm curious how is the "Check GW" message handled maintenance wise, especially if it occurs more than once?
Seems like something that needs fixing.
The FAC weight being higher isn't the critical case, if the FAC thinks the aircraft is heavier than it actually is the GD speed will just be faster.
So at least stall margin shouldn't be a problem.
(Making it less of a maintenance concern than the other way around)