As Ron Miller stated, the motorcycle-friendly camping trailers are heavy by themselves, not to mention adding additional weight when packing additional gear in the trailer.
Cargo trailers made by Hannigan, Bushtec, and Dauntless are all very good trailers. I have a Hannigan Europa model providing 27 cu ft of storage. It weighs 150 lbs empty and will easily carry 300-350 lbs of gear. Trust me, you do not have enough cargo capacity on the LT to carry the gear needed for an extended trip and have a modicom of comfort. By the time you are ready to load a tent, sleeping bags, clothing, a few potentially necessary tools and other necessities, you will see the benefit of a trailer. I could carry sufficient gear for an extended trip on the LT if I was the only traveler; with the wife along........no way.
The trailers mentioned above all have hydraulic shocks or some other method of dampening and minimizing the uncontrolled bouncing typical of non-dampened leaf-spring and coil-spring suspensions. The last thing you want behind you at 70mph is a trailer that bounces all over the place every time you hit a little bobble in the road.
I had no problem pulling 450-500 lbs of trailer and gear, two-up, on a 2200 mile trip at elevations ranging from 3000ft to 10,000ft. For safety's sake, I would not want to pull 600-700lbs. Your braking distance will be significantly increased, so, I would suggest you load the trailer and take it for a couple of test runs to get familiar with new braking requirements before you start your trip.
I don't know if any trailers offer surge or electric braking. If so, I would strongly consider that benefit when shopping for a trailer. Also, be sure your bike's brakes and rotors are in good condition and functioning well. The rear brake on the LT (at least the early models) is marginal but can be improved with EBC brake pads (several posts on this subject...just use the search function to find).