With stuck parts I usually use PB Blaster.
Heat is good for some situations such as rusted, difficult-to-remove bolts and nuts on farm equipment and vehicles. (repair guys sometimes call the acetylene torch the "fire wrench") The expansion of the nut can break the connection loose. Another option is to use dry ice to cool a connection. How either of these (or lube) would work in your situation is unknown. If the brass is forced into irregularities by deformation I can't imagine the heat option helping but who knows, it might if the bond is right on the edge of loosening.
I couldn't say if heat will damage your vise. Maybe someone else will chime in. Excessive heat can in fact do a lot of damage, such as changing the hardening/temper of steel as you mentioned, causing castings to crack, melting internal non-metallic parts, etc. I would not advise anyone to take a torch to something I haven't examined! I might do it myself, though, if it were mine, and take the risk. If so, I would disassemble everything possible first.
Pulling it out with a big bolt might be the easiest safe bet, using a nut to apply force. If that works, the threaded hole might be filled with a piece of threaded rod or a brass bolt.
If it were mine, I'd probably still cut it out then machine a new brass dog, replace the spring, etc. However, this may be an easier option for those of us who have milling equipment. Duh! Of course, if you'll settle for function over authenticity, you could make a new dog easier from hardwood. Dogwood would be perfect! I've also used Lignum Vitae for making various parts.
JKJ