On my coach the inverter powers only selected outlets like the kitchen, a couple in the living area, and the outlets for the TV's and such.
I have seen very few MH's with a converter since most do indeed have an inverter. Usually there is a breaker (or two) on the inverter or perhaps a separate breaker box of the inverter circuits. However if one of these are triped you would not have power to those circuits regardless.
You might grab some small 120 VAC deviece like a lamp, drop cord or ?? and just plug it into each outlet and test them while on inverter power. Does the TV work??