Gordon, I am afraid that the thermostatically controlled shower valve may not work very well in the RV. I have such a shower control in my home and it works very well. Because of the tankless heater in the RV, the shower fixture should have two separate controls; one for flow and another for hot and cold water mixing.
Some tankless heaters like the Girard Generation I and II, will not tolerate any cold water mixing.
As I said, it may not work and I may also be wrong on your situation. Good luck though and let us know how it works after the install.
Gale & Hank- 2012 Berkshire 390BH