Bob, (Sacramento, CA)
At nine years old, you may want to consider a new refrigerator altogether. Keep in mind, the cooling unit may be refurbished and charged properly, but all the other components; thermostats, burner, heating element, etc., are still nine years old. In my opinion, that's about the age when negative things begin happen with LP appliances. With a new purchase, don't forget, you’ll receive a new warranty as well as all new components.
But I would question the final troubleshooting deduction of that shop. If the refrigerator operates fine when level and stationary, the cooling unit is functioning properly. There may be other factors, such as accumulated heat, contributing to the malfunction while traveling. This Doc would recommend a second opinion given your provided information. It may simply need a good cleaning and some tweaking of the LP pressure to promote its effectiveness over the highway. Give this one some thought before committing.