Help to find the right stem/crown for a vintage Howdy Doody watch

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Hi all,

I'm working on a 1950s Bob Smith Howdy Doody children's character watch. The caseback is marked PATENT WATCH CO / SWISS MADE, and the movement itself is signed PATENT WATCH CO. - ONE (1) JEWEL - UNADJUSTED - SWISS MADE. It's an inexpensive Swiss pin-lever, and I can't find a caliber number anywhere on the plate. This is the moving-eyes variant.

The stem is missing completely. I have no crown either.

Three questions:
1. With no caliber number, is there any way to pin down what this movement actually is (ébauche maker, ligne size), or is measuring the only route?
2. What should I be measuring to order a generic stem or hunt the right donor - thread size, square, overall length, distance from the shoulder? Advice on how to take those measurements accurately would be very welcome, as would an honest opinion on whether chasing a donor movement is the more realistic path for a watch at this grade.
3. How to find the right crown for the stem?

Photos attached. Thanks in advance.