We have the same faucet in our Allegro Bay. Call Tiffin parts dept 256-356-0261 ex 5. Have your VIN handy. They should have a replacement, but you probably won't be able to get just the handle. Looks like the finish on yours is shot anyway.
FWIW, the finish on our shower faucet is going bad, too, and I'm looking to replace the faucet with one from Delta. They make one with an oval-shaped plate that should be wide enough to cover the 2 holes for the water pipes, and it also has separate temperature and volume controls. The model is the Carlisle
available in stainless steel finish that looks similar to brushed nickel. This faucet will require some plumbing skill to adapt it to work with a hand-held shower in an RV.