To run the A/C while on 12 volt power it would take about 150 amps at 12 volts and put a huge strain on both the belt and alternator. Better to use a 24 volt inverter for that purpose.
And A/C compressors are high torque devices, requiring pure sine wave inverter. MSW will not run it for long, and you will need a new compressor if you try it and the overload does not shut it off quickly enough.
So better to run the generator than to overload the alternator.
Yes you can replace the 600 watt inverter with a 2,000 one. However the wiring must be changed. right now, the main panel feeds 20 amps towards the inverter, then that in turn will feed say 3-4 outlets inside your RV. Not the A/C unit, not the electric water heater, and most important not the converter. It probably only feeds 3-4 outlets near the TV set.
I ran my microwave from my MSW inverter a couple of times, only for about 10 minutes max. I had to replace the microwave due to it's slow cooking times, from running on MSW power. The new microwave only runs on shore power or generator.
A good pure sine wave inverter will run about $1 per rated watt. Also a 2000 watt inverter will use upwards of 200 amps, requiring #000 wire. The 600 watt inverter probably has #8 wire and a 50 amp circuit breaker. At best maybe #6 wire and a 60 amp breaker. #2 wire is rated at 125 amps and the alternator output wire from the factory is probably #4 or smaller.
Good luck, and have fun camping,
Fred.