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High frequency arc starts help, but copper is pretty easy; it just sucks heat out. I would try a few practice runs on a similar sized wire first. If your brazing rods need flux, I would use a non corrosive flux given how close this is to electrical parts that might care...
FWIW...
Sorry!
Local welding shop? Easy to do with a torch, or better yet a TIG welder. It might have been silver soldered. (High melting point and high conductivity.)
I would be tempted to replace the wire if you can get to the other end...
All the best,
2Pbfeet