Winter tyres, whilst they won't do any harm won't really save you on ice anyway - they're more designed for colder temperatures and for grip on snow which won't really be an issue here, if you hit a proper patch of ice then without studs tyres aren't going to be doing much for you regardless of what season they're intended for.
I'd pretty much agree with others that an all season tyre would seem a better compromise for you - these will perform reasonably well in warm, cold, wet and dry conditions.
In the bad winter a couple of years ago I was driving down quiet roads (as in not gritted) on the way to work every day where the snow had melted to slush then refroze into solid ice with no issues on my vredenstein snowtrac 3's.