I used to be a millrite In my previous job. I had to heat up bearings, shafts, and other metal parts. While I never took a direct temperature of the part being heated I did see the parts at times change color. The color changes during the heating with a torch by the torch.
The hot lead probably would not affect the outside color of the barrel.
On a 200 year old gun I just wouldn't do it. Dealing with melted lead in the barrel sticking to the colder places in the barrel would also not be fun.