There are a number of good E rated LT tires that will handle that weight trailer. There are many reports of Chinese ST tire failures, but very few reports of LT tire failures.
Many go years without a failure, but I changed my STs for LTs after a year to minimize the possibility of a blowout and possible damage to the trailer or a wheel.
Good idea to also add a tire pressure monitoring system (TPMS). It will not prevent a blowout, but will quickly alert you if you have one or a low tire so you can pull off before further damage. Often there are reports of those who had a flat or blowout and did not know and continued to drive until someone pulled along side to alert them. By that time damage may have already occured.