Although I can't answer the "when" as to the date that the Carpathia picked up the "RMS" designation, it was not at all unusual for ships to be interchangeably known by the public as "SS" or "RMS" -- so this part of the puzzle isn't of great concern to me.
Typographically, the type face used is period, and the terminal period after the ship's name is also common. If this is a fake, it is a very clever fake indeed; not being a timepiece historian, I'll reserve my thoughts on the age of the clock, face, etc. save to say that it looks like it could well be the real thing.
That said, it makes the observer wonder if this was normal for Cunard to have a commonplace item like a clock emblazoned with the name of a ship. (I don't think that this was at all normal on White Star ships, for instance.) But if other clocks from contemporary Cunard ships can be found to have the ship's name on the face, then that boosts the claim that this might, indeed, come from the Carpathia.
It was somewhat normal for ships used for transport to have fittings stripped before venturing forth as a hospital ship or troop transport; what was the Carpathia sailing as when she was torpedoed? If it can be shown that her fittings were removed before service to the crown, then again, this is a good pointer to the clock being an original.
Are there any notices of the sale of her fittings after the war (as was the case with the sunken Britannic)? Again, this would be a pointer.
Could it have truly been the case that the clock was off of the ship being serviced? It's possible . . . but unfortunately, at over ninety years later (fifty when the clock came into your family), almost impossible to prove.
I'm fascinated by the clock, and hope for your sake that this is a real item as carried on the Carpathia. What a find this is, and I hope that your research finds the final answer.