I dropped by the mechanic’s today just in time to see him pulling my transmission apart to get to the component that caused me all this trouble. It was just a metal sleeve used by the reverse gear that stripped and lost teeth.
Naturally, it was located deep, deep within my car.
My mechanic gave me two options: take it apart enough just replace the sleeve, or completely rebuild the whole transmission while he was at it. I went with the latter.
So this little bugger and its missing teeth cost me $1,500.