Since you don't know the actual weight of your truck and trailer, the safest thing to do is go by the GVW as an estimate.
Your GVW is 16,000 lbs. A rough estimate of your hitch weight would be 20 percent of 16,000 lbs. or 3,200 lbs. Just as a starting point. Now you subtract that hitch weight of 3,200 lbs. from your GVW and you end up with 12,800 lbs. on the two axles. In real life it could be more or less. But that would end up with 6,400 lbs. on each axle.
The LT tires are rated to 3,042 or 6,084 lbs/axle. That puts you over the rating of the tires. And conceivably you could have more weight on a single tire since all positions/wheels rarely weigh the same.
And you're correct, most LT "E" rated tires are over their weight rating with your trailer. That's why it came with ST tires, lousy as they are they are rated to carry the weight.
If you want to go to a LT tire you're pretty much limited to "G" rated tires. There are a few, and then there is the Sailun S637 which is rated "For Trailer Use Only" and is now classified as an ST tire. And the Goodyear G614 which is pretty much a trailer tire even though it is designated an LT tire.