@GirthyChode I've been deep into the repair documents and engineering drawings for a week now tracing the inner workings. Fascinating machine. Unfortunately a replacement gear isn't an off the shelf item, so I'm going to have to keep an eye out for another one of these in a similar state so I can make one good one.
@kyle may be worth talking to a clock repair person. They have the materials, tools, and experience needed to make custom gears. It shouldn't be too expensive, either.