Sorry but using the lighter plug is your problem as you already discovered. Wire your inverter directly to the battery at the shortest distance possible with the heaviest wire the inverter will accept. Then run 110 volt extension cords to where you need the power.
I installed two inline transfer relays (link below) to power the circuits for the entertainment center and the outlets where we plug in laptops. That way those circuits are powered by shore power when plugged in and inverter when not, no action required by me it just works.
Transfer Relay