I finally got a chance to hitch the bike carrier, usually plugged into the short receiver in the back of the trailer, into the trailer hitch on the SUV. It works much better there; very stable and rigid. My plan was to use the bike carrier on the trailer during travel and then move it to the SUV when we reach our destination and want to go out and explore nearby cities and places of interest. Being on a bicycle gains you access to many areas where you couldn't normally go.
Based on our experiences today, this arrangement of the bike carrier on the SUV should work very well. I'm still not happy with the way the bike carrier works on the back of the trailer though; it sags too much. I've been taking weight off the hitch and relieving some of the sagging by tying the carrier up to the trailer's spare tire carrier. Not the perfect solution, but nothing has fallen off yet and I haven't pulled the spare off the back wall of the trailer yet either (thank God). Someday I'll have to take the rig to a local welder and get some structural support added to the arrangement.