I really must protest being on the West coast by the time I get up and on line the eastern coast has already post dozen of replies.
The Eagle did not have an inverter when we bought it. The Eagle had three 8 D for house, so I bought a cheap Xantrex 3000 inverter at that time for 750 bucks. Did not have time to wire so I plugged the shore power cord into it. We have used it a few times over the years. Mostly during the winter when the AC dock power goes out. Some people have trouble with the Xantrex, but I think its because they push it to the max, 2500 watts, for long periods of time.
Anyway I would have a gen set over a battery and inverter, but the inverter does come in handy once in a while. The trouble I have with an inverter is HOW it charge back the amps if you don’t run the main engine? You need a gen set!
Lastly I would install a separate charger and inverter.