Any 1500 watt generator should run any microwave. All micros are built to run on a 15 amp household circuit...what you have in your camper.
In fact, that's my test to see if my generator is functioning properly. I setup the genny outside my home kitchen and run an extension cord to the over-the-range micro. If the large microwave won't run properly, the generator is bad.
With no other loads (VERY IMPORTANT) the RV microwave draws far less power than a home micro. It should be a piece of cake for your generator to run that micro.
NOW...if you want to be absolutely thorough, pull the plug on your RV micro, run a 12/3 extension cord straight to the generator, and give it a try. That will rule out problems in your trailer. Now plug that extension cord into a 15 amp (not 20 amp) circuit in your house (shore power) directly and see what happens to rule out problems with the micro itself. Chances are it's the generator.
Other factors: When I setup, I run a 100', 12/3 extension cord from my 1500 watt generator to my (adapted) shore power cable. At 100 feet, 12/3 will easily handle transient 15 amp loads...e.g. the microwave at startup. With that extension cord in line, however, I can't RELIABLY run a 1500 watt hair dryer. It will occasionally trip the overload breaker, and in the case of my generator (a Generac), the only way to reset it is to shut down the generator then restart it. It was March in the Rocky Mountains when this happened, so the furnace may have been running, too....in which case, that would be a pretty serious overload.
But I NEVER have trouble running the micro or an espresso machine - one at a time - on my generator.
You'll get a raft of crap when you try this, but I'd bring it back to the seller and demand a replacement or refund. Explain that it doesn't deliver the advertised power (won't run a microwave oven), and you're not about to deal with warranty repairs or the manufacturer's nonsense. The seller can do that. Note that on your paperwork, the documents explicitly state that the seller is not responsible once you add fuel and oil, but most (e.g. Home Depot, Harbor Freight, Costco, and so on) will, in fact, reluctantly take it back. I returned...just returned...my first one to Costco, and bought the next one at Home Depot.
ANY GENERATOR can fail to perform up to standard...including a Honda. Get a new one, and do the home micro test on it. If it works out of the box, it will likely give you years of trouble-free service after that.
Look into the
Predator brand at Harbor Freight. Consumer Reports says they compare VERY favorably with Hondas for half the price. Champions are really good and inexpensive as well. But any single generator can be a dud. The inverter technology is the weak link in the chain. So if you have a Predator, just get another one. A close friend has a Honda and a Predator, and he loves the Predator...and it pairs with the Honda.