Check the manufacturer's label; I bet that the tire size will be ST185/80D13 indicating a BIAS tire. The tires presently on the brother's trailer are RADIAL (R13 instead of D13).
I had the same problem. When I purchased my "experienced" trailer, the dealer put brand new radial tires on it. When it came time to replace them, I couldn't find a D load range radial. I just put a set of Carlisle Sport Trail from Discount Tire. It cost me about $200 installed with road hazard warranty! I put 700 miles on them this weekend and I had no issues. They bounced a bit for the first 50 miles but then were smooth and solid. I made sure to have 65PSI in them.
Since bias tires are stiffer than radial, they react more to the road surface. Most notable was occasional wiggle from the ruts in the pavement, but not enough to be a problem.
For those who poo-poo Carlisle, the old tires were the Carlisle Radial Trail, lasted for 6 years, had half of the original tread left, and had almost 20,000 miles on them! I average 3,000 a year on the trailer; one of those years had over 5,000 miles in a single trip. So I felt that I had good life out of them. They were replaced because of age and one tire started to show damage in the sidewall and tread area.
2004 Toyota Tundra SR5 (V8, 4WD, TP, TRD)
2005 Fleetwod Allegance with axle flip
Honeywell 2000i Generator
Me, DW, DS, DD, & Chicken-Dog