Hello, Vermont. I guess it isn't just a local issue any more and your "state constitution" argument just fell flat.
The press release by Liberty Counsel does leave out a few details that are inconvenient for their case. Apparently the two women lived in Vermont for several years following the civil union, and when they split up the biological mother left the state to avoid having to comply with the Vermont court's ruling. There's more on the issue here:
http://www.freerepublic.com/focus/f-news/1678497/posts
I think precedent suggests that the state where a custody case is initiated (in this case, Vermont) is the state that has jurisdiction. However, it will be interesting if it goes to the SCOTUS. We will see what they think of the DOMA.