At 300 watts I think it would be hard to find an inverter that was not Mechanically quite.. Eletroincally a TRUE SINE WAVE is best.
Suggestion: Mount the inverter in an outside compartment next to the one the batteries are in (or if the batteries are on the tong of a trailer, just behind the tong) this way the whisper fan is OUTSIDE the living space..
Then run 120 volt wires from there to the TV.
Now the "electronically" comment.
MSW inverters use stair steps as their wave form.. If you ever visit a farm where their is a set of stairs going over a fence.. .These stairs are called a Stile. That is also the name for a pile of dung.
And the wave form of an MSW inverter.. This generates electronic noise and depending on how well it is filtered it can go all the way up to microwae (There is a piece of electronic test gear that does just this on purpose by the way,, I built one once).
A TRUE sine wave inverter, however, Sings as sweet and pure a note as my daughter's flute can make.. NO noise whatsoever. Just pure 60Hz. (That is between an A$ and a B, the 14th and 15th keys on a piano counting both white and black keys).
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times