Blue Moon is running software that is more efficient CPU wise, but less Database wise. what killed it on Elliquiy was the former. For the latter, AJAXchat operates mostly independently of the forum's database, so MyBB's rather horrific schema doesn't hurt it on Blue Moon.
I could setup a site with 'just' the AJAX chat, if people wanted, and that should mostly be fine. It can't be integrated with the database here, however, at least with SMF's verification functions, so... we'd have to consider how having the chat being 'more public' would affect things.
Alternately, if people want it back as the shoutbox, we can host another fundraiser. But I'm worried that would only buy a year or two of time (though hopefully new CMS would be finished enough before then).
Edit: And the reason I won't do a third-party solution is it would drop people out of SSL and be an overall privacy concern.