Mike,
I think you're right about them starting in 1937.
As to the repairing of yours, I think you may have a problem with that. Only because finding one with the parts your need may be difficult and/or having someone part with what they have even more so. They may be in the same boat as you.
The reproduction (re-pops) that will serve you right well and tits hard to tell the difference from the original. I think that the only readily obvious or noticeable difference would be the small brass pin protruding through the spring loaded keyhole flip cover/lid and the silver colored one on the re-pop.. That may have been corrected with the later ones but I don't know if it has or not.
They are relatively cheap (about 50 bucks) compared to finding an original that is complete and of show quality without having to have it re-chromed. A driver or show quality one will cost you much more. .

Good luck with repairing yours.
Best,
Charlie

Note: If you can find any GM cap that has the "A" on the eared plate on the bottom of it, then those parts may serve for repairing yours. I'm not sure about this but someone else may know better than I and let you know. Perhaps Gene, Ray, Chipper, et al.