You don't need or want a separate transfer switch. You are installing what we call a "whole house" inverter that will power the entire panel. Yes you can run the air conditioner, microwave, electric water heater, electric fridge if you like... not all at once and only until the battery is dead.
Typically you would not use the air conditioner and set the water and fridge to propane only. Any of these will drain the battery fairly quick. Use the microwave sparingly, better to use the propane cooktop when you can.
To fully utilize a 2500 watt inverter you should have 4+ batteries. For mostly light loads two batteries work for some people.
The batteries will recharge automatic as the inverter/charger has a charging function. Of course you will need an external source of power such as generator or utility. Just plug in the main cord. Do review the charger set up section in the owners manual.
The converter can be disconnected from 120v power and left in place. If the charging system should fail you could reconnect until the main inverter/charger could be repaired.
Once you have the manual etc in hand I think it will all just fall into place. You are on the right track.
I would try to avoid the DC cut off switch to minimize connections. Just remove the fuse or cable off the battery for service.
Manual calls for 1/0 wire and is larger than #1 AWG.
Custom cables available here