I don't know what Alex's plans are should he win both elections, but it is possible for somebody to be both a councillor and an MP (although it is unusual and byelections often follow).
But the cost of a council byelection is less than the cost of a parliamentary by-election, as we had when Tony Blair left the role of both PM and MP, he did not need to force a byelection. And then there was David Davis, who called a really pointless byelection and stood for re-election.