Distinguished SOTW member, musician, technician &
I do this with a dental micromotor, first suggested by Gordon in this forum actually. But it's like a dremel only in that it turns bits. It's a million times more precise, with better control. It is entirely possible to do this without damaging the post. Since starting to use this method I had to resort to other methods maybe once or twice.This might be possible if you went with a very fine Dremel etching bit, actually. But, again, it'd take a pretty experienced hand not to mess up the interior of the post hole (which is made of softer brass, of course).
By the way the rod screw stuck in the key hinge tube is the most common scenario, maybe about 90% of stuck rod screws I see are like that.