davidg, in this case what you want to do is add the inverter, straight off the batteries. Then add a transfer switch for the fridge circuit, it plugs into the bottom of the inverter.
Once the inverter & transfer switch is installed near the batteries. You find the power line for the fridge from the breaker panel. Disconnect that run, install a new power lead from that breaker to the 110VAC power-in side of the transfer switch, the line you disconnected, goes to the output of the transfer switch. Then the inside control switch for the inverter, install near the fridge and run the telephone style control to the inverter from that switch.
That was how our old factory installed residential fridge set-up was, before we removed it and installed a complete rig solar.
With this you keep the converter set-up like you have now and only had the inverter & fridge dedicated transfer switch.
Send me a PM and we can talk about an inverter & transfer switch that I kept from our solar upgrade.