If your trailer is truly 10,000lbs when you are towing, your tires are only holding 7500 to 8000lbs which should be well within the capabilities of a Load Range E tire.
Two things you don’t mention are the pressure you run and a TPMS. Air pressure in the tires needs to be checked as often as your fuel and oil as far as I’m concerned and should be at the maximum rating for the tire if the wheels allow it. A TPMS lets you do so constantly while on the road and is the best money you can invest.
A trip across a scale would be well worth your time and expense too.