Generator is a good backup to shore power. If you want a "backup to your backup" I would get FOUR 6V golf cart batteries, all new.
I would also suggest upgrading your inverter to a more modern inverter/charge/automatic transfer switch. With this setup you do not need a 12VDC converter, just a 12VDC distribution panel. Some even have an optional output to start a generator. There are several companies that make such a device, but I like
Samlex because of their simple installation.
Also, with a good house battery bank, you really need a
DC-DC battery charger, for charging while you are driving. READ THIS !