12V batts are usually 110 AH max. This is not good enough (in the long run), even if they are fully charged. Doesn't matter much what make, unless it's an AGM - those tend to drop less under load.
Coupled with some voltage drop due to thin cable from inverter, this could be enough to cause LVD.
Some cheesy 1000W inverter could struggle providing 900W.
So... updated homework to do:
1) find out the battery type, brand and AH rating
2) get somebody to check the cable size and length
3) find out inverter brand/specs, particularly - continuous power rating of inverter.
An alternative - add one more 12V battery and see what happens. You are going to need at least one more 12V anyway.
