I would like to replace the cartridge in my shower faucet handle with a pressure stabilizing one, so that running water in other parts of the house won't spike the temperature in the shower. The existing cartridge looks exactly like Moen model 1225. That replacement part doesn't say anything about being pressure stabilizing. Other Moen cartridges do though, for example this Posi-Temp one which is similar, but looks too large to fit in place of the 1225 model.
So basically I want the Posi-Temp functionality that can be used to replace a model 1225 unit. I have been to three different hardware stores without finding such a part, and the staff at each store has had no idea where I could find such a thing. I swear I did a replacement just like this some years ago, but I am really struggling to find the part I need. Any help would be much appreciated!