To be clear, I used the term "LC Cabinet Advisor" not to mean that such a position would hold the same privileges as a Cabinet Advisor (i.e. able to talk but not vote in Cabinet). I simply didn't have a good term for a voting member of the Cabinet who wasn't a Minister in the typical sense.

That said, I'm now realizing that there's actually no requirement that a Minister have an associated Ministry -- that's just customary. So there's no reason there couldn't be a Minister of Local Affairs position in the Cabinet that doesn't have a Ministry but nevertheless has a vote.