Oh I love Birdie! It makes an adorable nickname for Elizabeth or Bridget, but I guess those are too common for you.
Suggestions for Birdie as a middle...hmmm...
From your suggestions I like Cecilia Birdie and Alexandra Birdie.
I think a longer name (syllable wise) would work best. As for some suggestions (and sorry if I repeat anything)
I would try and avoid using a first name that makes it sound like you're describing a Bird. Like colour names, state names and other describing words like "Angelic Birdie" for example. It might come across as cheesy.
Out of the ones you have I love Charlotte Birdie! I think it sounds so chic.