SWAG- I'd bet a cup of coffee your axles are overloaded and flexing out of shape.
Check the vertical pitch of your trailer tires relative to the side of your camper without the bike and with. I'll bet they are pitching out at the bottoms.
In which case, the 'sway' you are getting is what we call 'hunting' as in the trailer is huntingnforma balanced line, but can't find it, so it moves back and forth looking for it.
Soft tires, under inflated tires, overloads, bad bearings, weak axles, back axle out of balance loading compared to front, etc will cause such behavior. Sometimes hunting is caused by a TV hitch is too high.
1k is a lot of weight, be careful.