stem puller for shower
Two options:


Find a screw that does fit the thread. You have the screw that holds the handle on, right? Take that screw to the hardware store and buy one that is threaded the same and 3-4 inches long. You can then drill a hole in a 2x4 for the screw (you'll probably need a washer) and use that 2x4 to pull the stem out.
Forget about changing the cartridge. Just open the back side of the wall and replace the whole assembly. If you have copper pipes, Shark-bite connectors make this a super easy fix nowadays.