You can go to 14” or even 15” tires with higher load ratings by buying bigger rims. Just measure your bolt pattern (probably 5 on 4 1/2”) or take one into a tire shop if you arent sure. Chances are your rims have a five bolt pattern that is also available on 14“ and 15” rims. Here are links to 14” and 15” rims that use the 5 on 4 1/2” bolt pattern:
https://www.etrailer.com/Trailer-Tires-and-Wheels/Dexstar/AM20315.html
https://www.etrailer.com/Trailer-Tires-and-Wheels/Dexstar/AM20422.html.
This is a link for measuring your bolt pattern:
https://www.etrailer.com/faq-wheel-bolt-pattern.aspx
This upgrade is a very good idea as most trailer manufactures use the cheapest tires with the lowest possible ratings as OE equipment, and premature tire failures on travel trailers are very common. These failures usually do some damage to the trailer (sometimes significant) and can also cause loss of control, when they self-destruct. Just look in your wheel wells - anything exposed or close to the tires can be ripped out by shredded rubber flailing around in there - plumbing, wiring, bodywork, etc.
So, on to tires. Most (almost all) trailer tires are made in china, and there have been countless failures due to inferior materials, design, and production practices. The only trailer tires made in the USA that I know of are Goodyear Endurance. I use them on my utility, horse, and equipment trailers and have never had a hint of problems. Either 14” or 15” will handle your weight load - just check that spec when buying tires. These are links to two 14” Goodyear Endurance tires that are 2050lb and 2200lb respectively.
Endurance® Trailer Tire | Goodyear Tires
Endurance® Trailer Tire | Goodyear Tires
So the last hurdle is fitting whatever you buy in your wheel wells. Measure them and take suspension movement into consideration. Seems to me that either of the 14” tires listed above (205 or 215) will be more than adequate and will give you a lot more confidence and peace of mind. And you can keep your 13” tires for spares although going this route you are must less likely to need one.