Hi Dave,
Unfortunately, which ever way you go, this is not going to be an easy repair. Assuming the rest of the barrel is in good condition (not needing a rebore, etc.), I would opt for a 'cold' repair. A 'hot' repair may well cause distortion or cracking. As I see it, the major problem is going to be holding the barrel without the aid of jigs whilst a new thread is cut. This may well involve screwcutting it as there is not much room for a conventional die and holder (if you can find one with the required thread!).
I guess you have already thoroughly cleaned the old thread - they often look truncated if they are loaded with rust from a previously ill fitting nut. Even threads this worn are not beyond recovery if you have a slightly undersize nut and the patience to sit there with a triangular file and carefully recut the thread using the new nut as a gauge.
Regards,
Eddie.