With it stock and no bed, its only noticeable back behind any vehicle with the popping noise from the lunette tongue.
The big weight items that can be swapped on the mil trailer are the axle with brakes, tires/wheels, the tongue, and the bed. The trailer frame is designed to carry weight and minimize torsional forces with a load. The shock crossmember is boxed for support. The other crossmembers are welded to support weight in the bed. The overall outer trailer frame is welded together.
The bed is rather large, but will hold vast amount of stuff. If you intend to keep the bed as a hauler, then I would keep the frame as designed. Lower it by putting the stock springs under axle and running smaller tires. You shoudn't need shocks, but you may later work them in. The stock shock mount is part of the axle tube. The bed is a drop down tail gate, so you could be putting that on the ground to load so watch how low you put the frame. The bed is wide so be prepared for that compared to the body width of the TJ.
As for weight with axle and tires swap will be couple 100#s if that. But all that depends on the replacement axle itself. Since the mil trailer comes with a parking brake, think you may want to keep that feature? Dexter axles have a parking brake option. Besides, with the parking brake, you can turn the trailer into a dump type by putting pins in the tongue front crossmember mounts. By driving the TJ backwards, it will tilt the trailer with the parking brake on and pins out of course.
Don't forget about the taillights. You can swap in normal 12v lamps easily and sell the mil lights. You can research for 12v lamps to put into the mil light housings. The wiring is mil spec as well as the connectors that are water proof.