I find it's not the helmet which is the problem. It's the gap between the helmet and ski goggles. What keeps me warm in the really cold is a windstopper beanie and two buffs. The beanie is tucked under the ski goggles.

A ski helmet is expensive. Too expensive even. I might consider an extra helmet if I got one on the cheap but you rarely find them where I live.

But I would still use one just for the actual mountain biking. The regular everyday cycling in the winter for me is not that risky. But of course we don't really get ice since it's constantly below freezing.