A stand alone unit is a lot easier to install (even if hard wired).
If you try to automate it, you need to have a way to disconnect the charger. Otherwise, it will draw battery power to run the inverter, which will run the battery charger, to charge the batteries and the cycle continues...unless you have come up with a perpetual motion machine, there will be losses all along the way and it will quickly deplete the battery bank.
You can kill the charger breaker manually but expect at some point to forget and wake up with a dead battery bank.
Keep in mind, AC loads are often higher wattage, so your pair of 6v golf cart batteries may not be up to the task.