When I want to burn a line, I use my old coping saw, with the blade turned backwards. Instead of a wire. It takes a little longer to get hot, maybe, but there is no chance I'll touch hot wire. I have a nice, safe way to hold a fine edge on the work.
Am I the only person doing this? Is there some problem with this approach I have not thought of? Aside from, it zaps the temper out of the saw and bends it some. You will not use that blade to saw again. But then, when was the last time you used a coping saw?
-Dan