You want the inverter for when you don't have shore power or gen. The inverter gets its 12v in from the battery bank and sends 120v along the same way as when it is on shore power or gen.
You can add a deck mount converter by placing it close to the F20 and run DC wires from the converter's output to the inverter's input. That lets the converter charge the batteries along the inverter's wires to the battery same as the F20 charger portion did.
You can plug the converter into any handy 120v receptacle or you might have to wire in a new receptacle by the F20's 120v input.
BUT!!!! Now, when on inverter for the 120v (not on shore power or gen) you must unplug that new converter so it won't be trying to feed itself in a loop. You will then be on battery for both your 12v and 120v as is standard for "off-grid."
(Before , with the F20, your charger part was cut off by the F20 when on inverter, but now that won't work so you have to do it yourself by unplugging the converter. Just remember to plug it back in when on-grid )