Ray, your eventual method is the one I would have suggested myself had I been here in March. The Dremel is good for this, and if you have the rounded heads, so are those mini hacksaws - the ones made of bent steel rod holding blades about 6" long. I have a couple of them, one always equipped with a coarse blade, and one with a fine one. I couldn’t imagine life without these two little treasures in my toolbox.
Oh, and I avoid Phillips screws altogether.
No self-respecting Canadian will go near the blamed things. Robertsons all the way.
If you’ve never tried a Robertson screw, then you’re in for a real treat, believe me. You’ll see… If I were an American I’d think I’d be leading some kind of consumer revolt for a universal switch from Phillips to Robertson’s.
On an application like that door, where rust may add to future problems, for sure I’d use screws that won’t rust. An extra dollar now, 'way fewer headaches later.
And if there’s anybody in yr family who paints over screwheads… well, let the consequence fit the crime. Get THEM to remove the screws. No allowance 'til the job’s done.