I suspect that the veneer is from the fridge manufacturer and not Beaver. Replacing it with sheet stainless is a great idea, though Marquis' probably evoke by design more woodwork than most.
Since our '06 Monterey has stainless Dometic fridge doors with no trim, I regret I can't advise on how yours goes together, Bobby. Perhaps you can get clamps on flat boards around the edges of the open door(s) and get the veneer glued back down. I tried heating old veneer once to get it laid back down, but it didn't stay for long. I think once the old glue lets go from ambient heat and humidity, and dries out after curling back, it needs new glue.
Joel