Short answer: yes, you can "just charge the battery" with that generator.
RV converter/chargers put out 45 amps, or 55 amps, or sometimes 65 amps to the battery. This is 45 x 12 = 540 watts, or 55 x 12 = 660 watts, or possibly 65 x 12 = 780 watts. Of course, there will be a certain inefficiency, plus things like your fridge control board, so unless you have one of the newer 65 amp converters, you will just be able to do do this by simply plugging the TT into the generator. But nobody better turn on a light or two.
There are two alternatives: one is to use the generator's 12 volt DC charger function. But if it has one, it's probably like 8 amps, and you won't like how slow that is. So you could spring for a 20 or 30 amp 120 volt charger to plug into the genny and do the job faster.
If you do overdraw the genny, it should have an overload shutoff on it, which will prevent damage to anything.
(yes, 1000 watts is about 9 amps AT 110 volts. It is also 83.3 amps at 12 volts, assuming 100% conversion efficiency. But there is no way you can get THAT.)
