It is possible, but probably not a good idea. Best thing to do would be to extend your xbox power button. And also, don't switch your "hot" use a relay instead. Switches don't have that high of a throughput. Also, maybe just extend your power switch that is already on the inverter.
And finally, screw the inverter all together, and get a 90 watt dc to dc ps for your Xbox, and just use that instead.