Sounds like your battery is well charged. You might try disconnecting 120V to the converter and see if it makes a difference. Leave your battery connected to the load center. Have you gotten up there on the roof in the last couple of years and removed the shroud and cleaned the unit? Having a dirty set of cooling fins will create all sorts of difficulties.
If you are measuring 108V when the AC is running, that should be safe. If you're tripping the load center breaker in the RV or on the pedestal with regularity, the AC is probably drawing too much current. A failing capacitor will also fubar the operation but you wouldn't be tripping the breaker.