Finally -- some findings to report!
I purchased a small number of 12V "flat" motors from eBay and AliExpress that had dimensions 30mm X 24mm X 18mm or smaller; the cost varied between $1.50 and $3.00 (USD) each. The shaft diameter of the motors is 2mm so to maintain compatibility with the Bühler gear. I also bought some replacement gears in case I couldn't remove the gear from the original motor. You can use a 0.4 modulus gear for interfacing with the original cog (4V motor interface), or a 0.5 modulus gear for the larger cog (12V motor interface); both have 9 teeth.
The gears were by far the trickiest parts to find as it was difficult to measure them and find a seller with gears of similar measurement. The 0.4M gear was not as readily-available as the 0.5M gear and I was only able to find the former in brass. The 0.5M gears are available in both brass and vinyl, and are the least expensive of all; a bag of 10 vinyl gears was less than $2.00 (USD) shipped -- the same cost for a single brass 0.4M gear. Before buying a gear, measure the diameter and length in mm of the original gear and compare with what the sellers claims, just to make sure the modulus isn't misreported.
Once you have your parts, it is really just a matter of milling the motor casing and adjusting the position of the gear on the motor shaft to ensure a snug fit in the Lego motor chassis. This is by far the most time consuming part, and was mostly trial and error for me.
Unexpectedly, the RPM of the new motor is much higher than the original 4V motor and thus produces A LOT of noise when run at a full 9V. I noticed that there is some minor vibration of all the gears in the chassis when using the original motor and so that vibration becomes amplified when run at a higher RPM.