I now understand how you're planning to wire this setup of yours. I think it will work. Mine is more manual. I use the Inverter when I want to be standalone. Then after nearly draining the battery, I then connect the batteries to a charger to fully charge it back.
So, in your case, you want 24x7 power from the Inverter. in that case, you need a Charger sized to keep the battery "charge" at the same time feeding your devices.
you need to research how to determine this power rating necessary to accomplish this. or crudely, i will attach an amp meter between the battery and inverter when you have all your devices plug in and determine the amps drawn. then choose charger based on that. (give it some headroom)