You don't state the wattage of your plug in inverter. My RV has two types of cigarette lighter plug in sockets. The one on the dash with the cigarette lighter element in it is a 10 amp socket. (provided by Ford) The outlets under the dash and in the stateroom (provided by Fleetwood) are a 20 amp sockets. Using my junk inverter (150 watt MSW) in the dash socket ended up damaging it and the 120 volt rated output would vary from +- 90 volts up to about 118 volts depending on the load on it until it failed completely. Using the 20 amp plug in sockets works much better.
Chum lee