The Winter Bash chatroom is not linked to a particular site, so there are no hats available for getting a star on New Year's Eve.
In my opinion, it should be linked to this site (i.e., Meta Stack Exchange).
No, it should not (at least not for the sole purpose of getting hats). Winter Bash is not specific to Meta Stack Exchange; it's something that affects all sites. Q&A about hats is centered here, yes, but it's on-topic on all per-site metas, too. If it was really tied to Meta Stack Exchange, it should be on Meta Stack Exchange's own chat server. At the moment, there's only one room on the main chat with Meta Stack Exchange as parent.
Stack Exchange chat has a tradition of starring 'Happy New Year' messages; it should not be too hard to find a willing audience :)