Proper name that uses Birdie as a nickname

My wife and I absolutely love the nickname Birdie for a little girl, but we are struggling mightily with a proper name to go with it. I want her to have a good proper name to fall back on as she grows up and maybe decides she doesn't want to be called Birdie anymore. Currently, the only name we are even considering is Bridget, but, even though it's a very pretty name, when I say it out loud to myself, it sounds harsh. Also, feedback from basically everyone we've suggested it to has been indifferent or outright negative. Can anyone help me brainstorm some other names?