Yes, just add the 15 amp adapter to the shore power cable and plug into the generator.
Yes it will charge the battery...just like being connected to shore power. The converter built into the camper handles the charging.
At the outlet of your generator, there is virtually no difference between the power offered on a campground pedestal or home outlet for a 15 amp circuit.
Warning: many will tell you that you can sneak by with a 2 KW (2000 watt peak/1600 watt continuous) inverter generator to run your
AC. That's nonsense. Here are the specs for an Atwood 13,500 BTU AC. STARTING POWER is 3800 watts. Can you get by with 3500 watts? Sure. Can you get by with 2000? Only in a fantasy world. It might run a few times, but you'll wreck the AC.
This comment may draw some debate, but the manufacturer's specs speak for themselves. And 13,500 BTUs don't change power requirements much from brand to brand. Bigger campers have 15,000 BTU AC units...they need even more power.
If you think you might want AC, you can pair two 2KW generators...and the good news is that you can decide to buy the second one later. If you just know you need the AC fairly often, you might prefer a single generator capable of 3500 watts or so. This avoids the headache of running two engines and maintaining two machines. But a 2 KW generator can be handled with one hand. A 3500 watt generator might weigh in near or above 100 pounds. Decisions, decisions.
Many here love Champions. They have a nice little 2 KW model that runs around $500 or so compared to a Honda/Yamaha at $900 to $1000.
If you boondock a lot, you'll appreciate some distance between you and your generator.
A 12/3 AWG x 100' lead cord is up to the job. It will easily deliver the 15 amps your generator delivers over that 100' distance. Next you'll need a good, case-hardened 1/4" chain (or cable lock equivalent) and a couple of top quality padlocks (also case hardened). Case hardening helps prevent sawing.
Almost nothing can resist the universal key. Lock your generator to a tree. The two locks give you flexibility and extended distance from the tree. Determined thieves can steal it, but opportunists will pass it by.
Next,
get one of these as a tent frame ridge pole, a couple of short bungees to hold the pole in place on the generator handle, and a 40 gallon trash bag (or equivalent and 3 tent stakes to make a rain-resistant cover for the generator. This will also make it less visible. In warm, dry weather, pull one flap back for cooling. The long ridge pole keeps both ends of the generator open for cooling. Use a big rock to hold down the 4th corner that gives access to your pull start. Inverter Generators hate rain, but this will let you run in the rain and disguise your generator. Or you could spend $50 for a purpose built generator "tent".
Last thought. How do you know you have a good generator and not a dud? Easy. Bring it home. Run a heavy extension cord into the kitchen Connect your big over the stove or counter-top microwave. It should be about 1200 watts or so. Your 2 KW generator should run that big microwave with ease. It is a good test device, because the microwave has high startup demands and it continuously draws near the generator's normal capacity (1200 watt microwave vs. 1600 watt continuous generator.)
If the microwave runs well, you're set. If not, bring the generator back for a replacement and test again. Don't take no for an answer on the replacement...some stores say you have to go for warranty repair. Nonsense. Junk out of the box is not a warranty issue. It's a replacement issue.
Pay with a credit card and threaten to refuse payment of they don't exchange. Easy.
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