Possible you have ruined the AC electric motor. Hum is OK for a few seconds, putting power to a motor that is not spinning can melt the copper coils wound in the motor - I thik, maybe.
It WILL NOT run off an inverter unless it is an inverter rated at over 3,000 watts, or over 20 amps. Then you would need a house of batteries, not just what you have on RV.
Air Cond plug is separated for a reason.
Run all your camper (except H2O heater) off battery.
Save AirC for generator or shoreline power. If you don't have, then you don't have AirC that day.
200 watt solar sounds like a lot but it is not a lot. It is enough for running the camper DC items some. It is enough to recharge and keep charged on sunny days.
It is not enough to run an inverter for any length of time.
Consider using only DC powered things. Save the inverter for when you have to have Alternating Current, and use it sparingly and not for long.
I use my inverter only to charge or run my laptop which only charges on Alt Current. Otherwise inverter is off. Inverter drains power even with no load.
Inverter is inefficient - it changes DC power to AC power and there is loss, then for some things you plug in a power adaptor which changes AC back to DC current.
Don't confuse AC air cond with AC alternating current. lol.
Air cond is usually written A/C,