A 1000 would be adequate, but might be taxed a bit, a 10 amp charge rate would be 120watts, a 20 amp rate would push that to 240 watts plus there's a conversion loss, so that's more likely to be 200 watts and 320 watts or more. Depending on your TV set it could draw a few hundred watts. So it should be able to charge and run the tv, but no more.
Look into the 2000 watt gensets, they're not much more and would give you a lot of buffer.
Now-2014 Sierra 346RETS 5er BUB
Then-2002 Keystone Springdale 286RLDS TT
Nights camped in 2014-28, 2015-127, 2016-10