As mentioned above if it only happens when pulling the trailer I would highly suspect it's the trailer. Any chance you could hook onto a different, but similar size, trailer and see if the problem is still there?
Generally a trailer tire being slightly out of balance (you mentioned 3-4 oz. and that is not extreme) will not be noticable. I would be looking at the run-out of the tire assembly on the trailer that may be caused by a bent hub or axle. Simply jack the trailer tires off the ground, spin them by hand, and see if they look wobbly in any direction.
As far as tongue weight and sway........you are only talking about 2,000 lb. fully loaded trailer. I doubt you could physically load it in a way to have too much tongue weight (unless it's some sort of homemade trailer with a weird tongue length or axle location).
I'm sure it's not an issue of too much tongue weight but rather too little tongue weight. Since it's a motorcycle trailer I'd suspect this is part of the issue, the bikes should be moved as far forward on the trailer as possible, since IIRC many of these trailers are built to keep most of the weight on the trailer axle (similar to a boat trailer). Not putting the bikes all the way forward could actually put you in a position where the tongue is trying to LIFT off the ball as you go down the road...which COULD be OP's problem. If, as they are cornering or hitting bumps, the weight is shifting OFF the tongue, then it is possible that the trailer could eventually come unhitched.
I've...actually had this happen once, or rather happen in a vehicle I was riding in. When I was a kid on a hunting trip with my dad and his friends to Montana, we stopped at the mouth of a mountain river for a short while. These guys liked to collect rocks as momentos of where they've been, and they piled all of these rocks (probably a good 300-400 lbs of rocks) in the very back of the trailer (a small 8 ft trailer made from the old box of a pickup truck). No one looked or checked the tongue weight, and over the course of nearly 2000 miles of almost constant driving (stopping only for fuel and food), the tongue lifting up on the ball finally snapped the catch off the tongue. The safety chains kept the trailer in check long enough to get down from highway speeds to about 25 mph, before those also gave and the trailer took a little tumble before coming to rest on its wheels.
That incident was enough to teach me at a very young age that loading and weight distribution is VERY important, and that what seems like an insignificant amount of weight added improperly to an already close-to-improperly loaded trailer can spell disaster.
Thankfully, no one was hurt, nothing in the trailer was damaged (only the side of the trailer was scratched/dented), and we were able to limp it to a nearby hardware store to make repairs for the final 60 miles home.
The rocks were left on the side of the road, thrown way off into the ditch. They recently re-did that section of highway, so I'll bet they're gone now.