It was incorrect for me to refer to all the brands I listed as "American" tires. Michelin was originally a French company but the tires are made in England, USA, etc. Bridgestone is a Japanese tire made ???
Let me say instead that I consider Michelin the best tire available regardless where the parent company is headquartered or where the specific tire is molded. Michelins will consistently produce 120k miles of service on a dually Ram costing just over $0. 10/mile. I consider Michelin, BF Goodrich, Uniroyal, General, Bridgestone, and Yokohama acceptable brands for truck/trailer use. My new Hitchhiker 5er came with Uniroyal Laredo LT235/85R16 LRE tires. They were not my first choice but are fine so far. I will run them until the carcasses are 4-5 years old then install Michelins.
Goodyear makes top quality high performance car tires, not so good truck/trailer tires. As I have written repeatedly, I wouldn't put Carlisle, Maxxi, Mission, Greenball, or other Chinese-made tires on any wheeled product heavier than or capable of being driven or pulled faster than my lawn tractor.
In answer to your question, no, it is probably not necessary that you change to 16" wheels and tires. Your signature doesn't tell us what trailer you own so I can't offer specific suggestions. If I had an RV trailer with 15" tires I would buy the best commercial grade or SUV application brand name tire available from the list above.
I installed a set of Michelin 15" SUV tires on a 9,000 lb. Avion TT several years ago and put 60k miles on them, including a problem free trip to Alaska before I sold the trailer. BF Goodrich makes a good commercial grade 15" tire IIRC.