I'm adding one more opinion that you should heat it.

Additionally, It may help to mangle the shower screen and force it to rotate so that you know it isn't adding to the static friction holding the screw.

Lastly, if you mangle the shower screen enough, you can bend it completely away from the brew head so that it is shaped like a taco shell around the stuck screw. Then vise grips can grab the screw and taco as one complete piece and torque the screw out of the brewhead.
Sacrifice the screw and shower to save the brewhead.