Unfortunately, for the time being, you'll need a SO account to log in to the Winter Bash site.
Winter Bash is its own site and not part of the Stack Exchange network, so it is subject to most of the same limitations as any other site on the internet. Logging into Winter Bash really means getting a Stack Exchange OAuth token via Stack Overflow, similar to the way logging into Science Fiction & Fantasy can involve getting a Google OAuth token by logging into Google.
In the ideal scenario, we would show you a non-SO branded login page, accept your SE credentials, and forward you on to your destination without requiring you to fully log into the site doing the login (i.e., creating an account). But our login process is far from ideal and there's no immediate workaround.
I've filed a bug internally, and our auth SME is thinking about it, but I'd say it's unlikely this'll get fixed in the next 3 weeks. The best option I can give you would be to sign up for Stack Overflow (log in with your network user name and password and click Join) and then hide the site from your profile. You will still have a Stack Overflow account, but it shouldn't leak through into your profile on other sites:
Hiding a community means other users will not see it in your Communities list or Top Network Posts. It does not, however, mean your activity is private.