Flux? NEVER use flux on electrical
connections because it is corrosive. Solder sold for electrical work has rosin core
(or nothing at all); rosin is NOT corrosive. So not soldering because of flux is a bit irrelevant. Just use the right stuff.
Personally I would solder the two wires together (with rosin-core solder) and paint
(or dip) with liquid vinyl. Don't use a 235W Weller soldering gun (or an old-fashioned soldering iron); try a 25W-35W soldering pencil. Putting heat-shrink over the liquid vinyl is overkill for this size wire, but if belt and braces is your style then go for it.
Work hardening of the copper at the edge of the solder joint is only a problem with vibration, so secure the joint so it doesn't move. The liquid vinyl or heat shrink should do that. Or fasten it down. Easy.