A winter road bike would work. The biggest things I'd want to consider are gears, do you need them? Mudguards, you need them! Rack or Back? Strong wheels are important if you're riding in traffic and the dark as it's not always easy to avoid things.
You could do it on a slicked up mountain bike but I personally would be going cyclocross, touring, audax, winter road bike for fast reliable options.
I like on-ones and there's on in the planet x clearance so I'd consider and uncle John or kaffenback.
12 miles isn't that far, I'd have thought 45 minutes would be easily achievable on a mountain bike so you don't need a proper race bike.