I was curious so I went out to the MH and plugged the TV into the adapter, then the adapter into my 300w PSW inverter. The TV was tuned to a local station with the sound at a normal listening level.
The TV running with the adapter drew 0.14A @ 120v, 16.8W. I couldn't measure the current out of the adapter. The adapter says it is rated at 3A at 12v, 36w.
Running off the inverter and the adapter it drew 2.0A @ 13.2v, 26.4w.
These measurements were done with my clamp-on ammeter.
Doesn't look to me that a 300w inverter is required.