I'm a bit confused by some of this, so I will start fresh.
It is very do-able. To me, the easiest way is to get an inverter of the size you choose with a BUILT IN ATS (Auto Transfer Switch).
1. To wire it, remove the circuit (s) you wish to run with the inverter from the main circuit board.
2. Use the proper size breaker installed in the main breaker box, and connect this to the "AC IN" on the inverter.
3. Take the inverter output to a new sub box, powered from the inverter, with breaker(s) to power the circuits removed from the main panel in step 1.
4. Run HEAVY CABLE from the inverter to the battery or fuse panel. The 12V cables should connect to all of the following: Battery bank; 12V fuse panel; converter / charger; inverter input.
With this setup, when 120V is present (shore power), the inverter receives this power and the ATS passes this power through to the sub box. No battery power is used in this setup.
When no 120V power is present, the ATS flips to take battery power and invert / convert to 120VAC, which is sent to the sub box.
No specific action is needed on your part. The ATS handles it for you.
A couple notes: Only power needed circuits. Make sure A/C and water heater are NOT on the inverter. Do not use the inverter for electric heaters.
Comments above on the microwave are very accurate.
Make sure the converter is NOT on the inverter. This sets up a loop where the inverter powers the converter to charge the battery due to power being used by the inverter.