Joe,
I think you're mixing "inverter" with "converter". Your onboard converter uses shore power to charge batteries and provide 12 V power. An inverter takes 12 V power from batteries and provides 120 V AC.
There are no issues about leaving a 2 x 6 V battery set independent of the trailer's original 12 V battery. The question arises, why? If you wire the new 6 V's into the RV's charging circuit, when you're on shore power they get charged. Also, connecting the 6 V set into the existing 12 V power distribution, the whole battery bank nets you more power.
If you want to use the 6V's as an emergency backup, wire all the batteries together and use a main battery disconnects switch that has four positions: Battery 1, battery 2, Battery 1&2, off.