Your brother who lives in Christchurch is an electrician, while some other brother lives elsewhere and may have another occupation.
Your brother, who lives in Christchurch, may be the only brother you have, and this would indeed be the construction were that the case, but your sentence could just as easily be followed by:
And my other brother, who lives in Wellington, is a plumbing contractor.
As far as I know, there is no rule of English grammar that insists on one's thinking of every male sibling when saying something about one in particular.
The restrictive versus non-restrictive rule according to how many siblings one has is also valid, though often not observed, with appositives:
This is my brother, Mike. [you have only one brother, and that's Mike.]
This is my brother Mike and his wife, Sandra. [You have at least one other brother, but Mike isn't a bigamist, so his wife gets a comma.]