Since it isn't dad's pipe or a priceless something-pipe, try the mud thing. If it doesn't work, pitch it.

If the patch holds and the wood no longer darkens after a while, try a little sanding, alcohol (suggested above) to reduce the spot and restain and refinish. All told you're taking a few hours. The effort put forth is only for you. If it is some saleable name (if undamaged), it won't be worth much with a spot (or not) and a burnout repair. IIf the pipe was a cheapie to begin with the value of the pipe and the effort to repair it is whatever you think it's worth to you (aesthetically).

I can't explain why I took acouple of hours (I had to have been worth at least $100/hr back when I could still think ) out of my life to scrape, scour, mud, sand (inside and out) and wax a $7.95 MM General. WTH? I just liked THAT pipe. Still do.