I had a 34' heavy frame triple axle bumper pull as well. Pulled like a tuned arrow... triple axle and properly balanced. The height of it was nice and low. But I was also pulling it with a one ton dually. The frame on an enclosed car hauler (big one) vs a camper is night and day difference.
There are a lot of factors for his sway situation. One is the camper pulls like a wet noodle and send a frequency tat gets amplified at his rear tires. The balance of his trailer being a toy hauler also helps NONE.
This is the best summary of the issue. Its the trailer design. And its poor enough that it's got him questioning the truck.
If its as bad as the op says, which I believe, id be looking to get rid of it. Add to that "door not closing for the 6th time" and you can imagine that trailer moving like a snake as it bends and weaves like a slalom.
I think until you get a different trailer you'll be sparring your truck against your trailer.