If you want to power the existing outlets you will need a small transfer switch.
https://www.donrowe.com/KISAE-TS15A-Automatic-Transfer-Switch $50Switch is easy. Remove the romex of your choice of branch circuit from the breaker panel and connect to the switch OUTPUT. New piece of romex and reconnect to the breaker panel and connect to the INPUT. Plug the switch into the inverter and you are good to go.
If the fridge is on the circuit you choose you will need to set it to propane only or it will run down your battery very fast.
Inverter close to the battery is best but then you are pulling wire as needed. For just small loads you might be able to mount the inverter behind the breaker panel and supply 12v withe the main battery charging wire. This would put all connections near the breaker panel and just make the connections. No pulling wire through the floor or walls.