Our furniture in the travel trailer isn't strapped down and for the most part it stays put. However, no one rides in the trailer.
If the couch has seat belts, I would want it fastened securely. Our bounder had a metal bracket with a plastic screw on handle to hold the chair in place and a lever that locked to keep it from rocking during travel. The movable chair AFAIK is never an "approved" seat while in motion. I would personally want the chair firmly fixed in place while underway. Most of the time it won't move, in a panic stop it could start sliding.