Thanks for all the good advice. I got a puller, and (at first) bought a Home-Depot-generic 1222 cartridge. It was only after I got home that I read the reviews for that cartridge, which were overwhelmingly negative - and most of them finished with "then I bought the Moen cartridge and it worked great". So then another trip to Home Depot to swap it out...
I did find that even with the puller I needed more torque to crank out the 22-year-old cartridge - see the photo.
At certain points of the extraction I had to grab the pipe that the cartridge was sitting in to counteract the pulling - all the plumbing behind the shower enclosure was not fastened to a 2x4 and was just floating around back there, so all that pulling otherwise could have put the plumbing and the fiberglass wall at risk.
All's well that ends well. Hopefully I am good for another 22 years. Thanks again for all your help!
- Steve