Bob, Some points to ponder:
1) Consider a shore power pass-thru type inverter with ATS (auto transfer switch), or combo inverter *charger)…Pay attention to the inverters surge rating and surge duration – compare this to your air conditioner’s LRA (compressor locked rotor amp) spec’s in owners manual.
2) Plan to install your new inverter as close as possible to the batteries (6-7’ round trip cable length preferred) and I’d recommend AWG 2 aught (002) gauge to reduce voltage sag (e.g. avoid battery bottlenecks and low voltage alarms).
3) Consider adding a sub-panel to segregate the [battery sourced] inverter from powering up the OEM converter-charger, hot water heater, and the **refer
4) Consider increasing battery capacity and/or consider switching to lithium (LiFePo4).
5) Review your microwave oven’s ‘gross’ wattage spec (found in owners manual).
Consider downsizing to a more economical 900-1100 gross rating – they save some power and work very well indeed!
* if switching to lithium (LFP), ensure that inverter-charger combo is LFP capable (may require a few other charging considerations).
** some refers ‘sense’ AC, then auto-switch from LPG