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.
2018 Forrest River Salem Hemisphere 282RK - 2017 RAM 1500 TV
Previous RVs and TOADS
2004 Fleetwood Bounder 32W on WH W20
2000 Four Winds 5000 21RB
1986 27' Allegro
TOADS
2005 Ford Ranger XLT 2WD
2004 Suzuki Aerio
1988 Chevrolet Sprint