Been using a honda 2k and knockoff 2k for about 15 years. Got my first knockoff prior to the honda. Always had trouble starting it, but it would run great once started. This at the time was one of the only knockoffs being manufactured, it wasn't a japanese brand. Since then I have also gotten a super cheap, $79 1k harbor freight generator that I wanted as an emergency backup for my 2k generator. I took it when we were sailing from Bahamas back north; it made it back but barely. But I think there are a bunch of knockoffs today that are pretty good, ranging from brand names like Yamaha, which cost pretty much the same as the honda, down to store brand OEMs at 1/3 the price.
I am a big project farm advocate, his info is great, but if all you want the generator on the boat for is to connect to your charger, sinewave accuracy under load really isn't important. If you are plugging an AC motor into the generator, it is very important.
Also, I think if you can afford the price difference, I would spend the money and go for the 2k vs the 1k. The 2k is like a 'real North American' circuit, you get ~15A. The 2k is not much heavier. The 3k is much heavier.
I don't see why any 1k generator would have an issue with a 50A charger, that is only 600W at 12V. If you have a 24V bank, then yeah, it won't keep up.
AFAIK, all small engines will have their carbs 'gummed' up by gas you get at the station, even premium. Either put stabil in and change every 6 months, or do that and when you use the generator, as others here have said, run the carb dry. You can also get fuel wo ethanol at small airports as private prop planes won't put ethanol gas in. Home depot also sells ethanol free gas in small cans, though I think it is like 4x the cost per gal. But if you just need to fill your weed wacker once a week, not a bad way to go. The guy who sold me my outboard told me that he tells all his clients to change their gas on a monthly basis to prevent it killing the carb. Not sure if you put stabil in if you need to do it that frequently, but he is saying not to leave it very long.
I will never sleep with the generator on! I have seen some aftermarket addons that will take the exhaust with a pipe into the water. Possibly with that, if it was on my swim platform, I was on a mooring/anchor so I know I am facing into the wind, and I just tested my CO detectors I might consider it. But still probably not. Plus just not nice for others in the anchorage to listen to.
My 2c of using these portable generators for the past 15 years.