Reset the inverter and things came back on?
Unplug dish and TV.
Reset the inverter.
Install (temporarily) some small AC device - like a radio, cell phone charger, something that you can tell is "on" but doesn't draw much power.
See if it stays on under inverter power. Trying to see if the TV/Dish are drawing too much power for the inverter.
Does it make any difference if the inverter is OFF and you're plugged into shore power?