I have a piece of moulding on an appliance that is held on by a long, thin screw with a hex head. The screw is perhaps 5" long and perhaps 1/8" diameter (modestly heavier than a wire coat hanger).

The overall dimension of the channel is slightly larger, but not substantially so.

When attempting to remove the moulding to replace a light bulb I noticed a lot of resistance on the screw. I applied penetrating oil, waited, jiggled, repeated a few times and finally managed to snap the head of the screw off along with perhaps 1/2" of the shaft.

I now realize that the screw is seized at the bottom as well.

What sort of bit could I use in a cordless 18V drill to remove this safely?

Is there another technique that would be applicable given that I cannot replace the moulding and thus must make a serious attempt not to damage it?