There are multiple issues that might add up. First TV's and "House" type radios Sat Receivers, DVR's and the like (Any device which is turned on and off via a REMOTE control) all suck some power. Just a few watts each but you only have 200 watts overhead.
Second. What is the shut down. The inverter at 800 watts is sucking close to 80 amps off the batteries. if you don't have enough battery. if the batteries are getting "long in the tooth" if the connections are even slightly dirty or if the wires are too small 10 volts happens and most inverters are long since in ERROR mode (like at 11.5).
Finally there are two ways to rate electrical power sources. Some companies (Honda for example on their inverter generators) a EU-2000i is 2000 watts PEAK. sustained it's not that good. The Xantrex inverter I ordered. per the ads and ratings, is 2,000 watts forever double that for so many seconds.
The most likely issue is wire size on the 12 volt side of things.
