Grab the screw head with a pair of dykes push the handles toward the cover to leverage the screw out. At the same time as you apply pressure attempt to remove the screw with your screwdriver.
I like this idea. Unfortunately the only tools I have are the ones I will find in my mother's tool bag, but perhaps there will be a pair of dykes down there at the bottom.
The dyke trick can be done with nearly any metal object, it just gets a bit wonky when you can only hit one side. With something like a butter knife.