HI guys,
I ended up buying this auto choke from Taiwan for $22. The pictures looked just like the $5 parts, but it was labeled as being for the A125. It took about two weeks to arrive. It looks exactly like my old one, down to the color of the connector. The install was pretty straightforward. A little hard to get to, but pretty straightforward. It seems to have fixed my problem.
Interestingly, it didn't completely fix my problem until I replaced the rectifier, which was also going bad. The new auto choke worked better, but not perfectly. It works correctly now, with the new rectifier installed. My thought is that the bad rectifier wasn't giving the auto choke enough voltage to heat up completely.