I took some measurements a while ago from different size inverters and I thought we had it all sorted out that the "idle" draw means nothing for this question, which has to do with when loaded. The load is not "added" to the idle.
What counts is the efficiency curve of the inverter where they seem to peak at somewhere over half their watts ratings. If you max it out with load or underload you get poor efficiency, which translates into higher amp draw to run the same size load.
So to draw the fewest amps on a known load, you want that load to be say 3/4 of the inverter's watts rating.
TVs have a variety of load depending on your picture mode setting, which way overpowers any consideration of inverter size choice. Say you decide to go for the brightest picture on your 32" LED and that makes it 100w. (dull picture might be 60w)
So you would want 100 to be 3/4 of X so that makes X 133w as ideal. If you choose 2000 vs 1000 then obviously the 1000 is closer to 133 but in real life the "loss" in amps between the 1000 and 2000 won't be very much as both are far off that 133.
Which brings you to the time factor. The battery loses AH not amps. If the TV is on all day, a small "loss" can matter, but if you just watch the news and weather and shut it off, you can't get many AH no matter what the amps come to.