UPDATE #2

Got impatient. Managed to mangle the two remaining screw heads. They didn't budge.

Classic me.

Did you heat them up first?

Yeah, I gave 'em a good two minutes with a high-powered blow dryer, but no dice after the first one came out.

There's still technically enough of a slot left that a correctly-sized bit for my impact driver *might* be able to jar them loose, but I'm in the crappy position of having to order stuff online and wait several days for each new technique I attempt (in the process acquiring a bunch of tools I already own in another city!)

I was guardedly optimistic I'd be able to tackle this project with just a screwdriver, some Cafiza and a bit of touch-up paint, but I'm afraid it's turning into a bigger source of aggravation than I was hoping for.