I am trying to replace my see through Moen plastic shower handle and the Phillips head screw that holds it on won't budge. How can I get it off?

Purchase or obtain a 3/8" drive speed handle wrench and an #2 anti-cam out screwdriver bit.
The anti-cam out screwdriver bit has little ridges to help stop the bit from slipping in the screw head. The speed handle allows you to exert the maximum pressure into the screw at the same time you have maximum turning power.

Cheap speed handle about $10. Screwdriver bit about $2.
If you have trouble finding anti-cam out screwdriver bit, look for Snap On tool truck.

Why not simply drill the screw out and replace with a larger self tapping screw? Sound like a plan? Good luck, Tom