Heat not load causes tire failure. Now if you are carrying a heavy load at high speed you will be cause the tires to overheat, so load does enter into it.
You may or may not need a higher load rated tire. You need better heat dissipation. A larger load rating may not solve your heat buildup problem, if it is caused by the speed you are driving. ST tires cannot handle heat very well. They have thick side walls and retain heat.
Go scale your trailer and then find a LT tire with a load range for your trailers weight. IF none are available for your weight then look at 15 inch tires. LT tires have a much better ability to dissipate heat and can be driven faster without a heat buildup.