Welcome to iRV2.
Me thinks you're over estimating the power supply the 400 watt , inverter is capable of. If your basic outlets include anything but a flat screen TV or a digital alarm clock, it won't last long, if you plug the system in and the fridge is set to auto , and the control board tries to select 120v power, it will trip the inverter's C/Breaker or ; worst case burn it out.
The second thing , now that I think about it , would be hooking inverter power up to the transfer switch , would require you shut down the converter/charge before you hook the inverter ; or you would create a loop, the inverter drawing off the batteries producing 120v, the converter/charger trying to produce 12v to charge the batteries at the same time .
The factory 400 watt inverter in my coach , was wired into dedicated outlets, for the television and one at the passengers feet to charge a lap top or cell phone while the coach is being driven. It also had a manual by-pass to hook those outlets direct to the coach's 120V when the RV was on shore power.
In all this , we don't know what kind of RV you're trying this out on and what you have available in the way of 12v battery power.
EDIT: And I see another member posted while I was away typing. Boy do I type slow.