I did the same thing and tried to use a small 350 watt pure sine wave inverter in our rigs entertainment center where a DC socket was located. No dice as the inverter squealed mightily due to the small diameter wiring used by the manufacturer. I was not surprised by this knowing how RV manufacturers cheap out wherever they can, I still use the DC socket for a plug in DC voltmeter.
My choice was to either locate inverter close to battery and then run extension from inverter to entertainment center, or run a much larger diameter wire directly from battery bank to feed inverter in entertainment center, chose the latter and now PSW inverter easily runs our HDTV, HD satellite, soundbar & subwoofer, HD gaming system, and for charging portable electronic items.