Came home from work and took out the screw holding the handle mount in place. Just like the handle before it, the mount wouldn't come off - it was stuck to the cartridge. Hit it with the PB Blaster a couple of times (I don't think this project would've been possible without that stuff - however I read it isn't good for silicone o-rings so I kept it away from the new parts) and eventually I was able to muscle it off.
Took the screws out of the face plate and removed it with no trouble. Removed the U-clip that holds the cartridge in place. Used the plastic tool included with the new cartridge to break the seal on the old cartridge by rotating it, then was able to pull it out with a regular set of pliers. I bought a cartridge puller but didn't end up needing it.
Cleaned out the valve with a toothbrush and some vinegar, rinsed it, then put silicon grease on the new cartridge's o-rings and popped it in. Pushed it back into place, put the U-clip back on, and away I went. Handle is a lot smoother than it was, no leaks so far.
Funnily enough, after I was almost done, Moen called me back. I had sent them an email asking what the difference was between the 1200 and 1225 cartridge. They said they were interchangeable. I give Moen high marks for customer service. However, their instructions for changing a cartridge say that an expert can do it in 15 minute and a novice can do it in an hour. Maybe if you were replacing a cartridge on a brand new installation...but with all the corrosion and crap this took me more like 3 hours, not counting the multiple trips to home depot...
For reference, this is a good exploded diagram of the system. My handle mount was the pre-2003 variant.
Thanks to everyone here for your suggestions.