Your batteries no longer charge to full charge. What do your batteries read 12 hours after charging and removing shore-power?
Reading the meter with RV plugged in just tells you charging is taking place. The battery could be no where near full charge. A 1000 watt inverter most likely uses 0.6 amps each hour for over-head. Did you use a clamp-on meter to check the amps off the cable between inverter and the battery? That would have given you the amps expected to be used each hour.
My 1000 watt inverter requires 6 gauge cable (about 2 to 3 feet). Anything smaller will cause a larger "load drop" on voltage and premature "low battery" shut-down.
I suspect your battery may be sulfated and recharging to only 50 to 60% new capacity.