can it be done.. yes..
but for the diagram you have drawn, you Need a whole house inverter, something that will carry 30amps (not supply..just pass thru) or more of 120vac
and you will need a switch/Circuit breaker to turn off the converter (remembering to do so is the big problem)
actually much better to parallel the converter and charger 120v input, so that the converter is automatically off when there is NO shore power
people usually use a "break out sub panel" to power the needed circuit from the inverter
everything has power when connected to shore
with out shore power, only the inverter circuit has 120v power
