One of Mary’s quilting group arrived with a machine in dire need of cleaning and oiling. These screws hold the throat plate in place:
They’re standing in a pair of threaded brass inserts (found in the benchtop litter) to show off their tops.
The left screw came out easily, although a few licks with a fine file eased the slot corners.
The one on the right, however, was firmly jammed in place, with the crappy little Kenmore sewing machine screwdriver causing the goobering. I deployed my Brownell’s Gunsmith Screwdriver Bits, applied slightly less force than would ordinarily call for an overnight penetrating oil session, got the screw out, and cleaned it up:
A dot of oil on the threads should keep it happy for the foreseeable future.
The Smell of Molten Projects in the Morning 
The Brownells bits look like a great set to keep in reserve for just such occasions. I have some Chapman bits myself for similar situations.
I have a couple of top quality “L” screwdrivers for tight spaces like you have, saved me many problems over the years when I was a machinist nd mechanic. It’s the selection right tool for the right job that is the secret.
There’s a small right-angle ratchet driver tucked away for those special occasions arising once every two or three blue moons!