The battery is a group 48PG, 730 CCA and the tag says 20 hr capacity is 70Ah and load test 365.
Going with the 70Ah number, at 1080W / 12.8V / 0.85 you again get 99.2A. Which for a
healthy lead acid battery shouldn't introduce too much voltage drop, but if it's worn/used/abused due to poor charging conditions, you could get quite a lot of voltage drop. When voltage drop happens, the amps required will go up...which can cause even more voltage drop (and more rise in required amp)...it's vicious cycle. A repaired or new battery would be the fix in this situation. And with such a large inverter, I would double or triple the aux battery capacity, so 200-300Ah would be good.
There are two MAJOR advantages of lithium batteries over lead acid:
1) far less voltage drop. Lithium can take much more amp draw and the voltage will drop only a little and no vicious cycle like lead acid.
2) far less finicky on how they are charged. Lead acid wants to be charged to 100%, and stay there, all the time. They dislike being discharged and then not fully charged back up (hard to get a full charge on alternator alone). Lithium will take 1A of charge or whatever the max charging rate (C rate), which for a 70Ah lithium would be usually a max of 35A (0.5C rate), but it will be totally happy with a rate less than that. And lithium prefers to not be charged to 100% all the time, so the short/incomplete charge cycles from an alternator (and one that varies in output due to up and down of the RPM's of the engine) are just a-ok with lithium. Only catch is lithium cannot be easily or safely charged directly from the alternator (could burn the alternator out) so you need an appropriate DC-DC charger,
like this one.
Probably not the sort of thing a gov't agency, who are famous for being stuck in the past, is going to pony up for, though. Even though it could save them money in the long run (2000-4000 cycles of lithium vs 200-400 cycles of lead acid and could be 100 cycles or less if battery is abused/improperly charged regularly).