please excuse my edit
My grandmother refers to all of her daughters as something-bird. ([name]Nanny[/name]-[name]Bird[/name], [name]Susie[/name]-[name]Bird[/name], [name]Katie[/name]-[name]Bird[/name]). I imagine [name]Birdie[/name] would follow pretty logically from that, which means you can use just about any name…kind of like a little boy who ends up being called [name]Buddy[/name]. Lady [name]Bird[/name] [name]Johnson[/name]'s actual name was [name]Claudia[/name].
(This was actually my plan for my own daughter, but now I have a chubby cheeked 3 month old who it’s hard to imagine as a “[name]Bird[/name]”. Maybe when she’s a little older?)
I was going to suggest giving her a bird name and calling her birdie 
[name]How[/name] about these names which have sounds that could make [name]Birdie[/name]:
Abira – strong, powerful, noble
Bira – fortress
[name]Birgit[/name] – exalted one – I like it better than [name]Bridget[/name]
[name]Birgitta[/name]
Birkita – woman of great strength
Kabira – one who is powerful
[name]Sabira[/name] – patient
Or things with the Ber sounds:
[name]Alberta[/name]
[name]Amber[/name]
Berdine
[name]Berenice[/name]
[name]Bernadette[/name]
[name]Bernice[/name]
Bertila
[name]Ember[/name]
[name]Kimberley[/name]
[name]Roberta[/name]
Or names that mean bird:
[name]Aviana[/name]
[name]Avis[/name]
[name]Mavis[/name]
[name]Meena[/name]
[name]Merel[/name]
[name]Palila[/name]
[name]Sephora[/name]
[name]Tzipora[/name]/[name]Zippora[/name]
I’ve seen the name Birdine, which I think is cute. Also, though your husband probably won’t go for them, there are the names Songbird or Bluebird. Here, if you go to this link you will find a list of girl’s names that contain “bir.” That may give you some options.
Of course it would help if I added the link: http://www.babynamespedia.com/contain/f/bir
I think [name]Bridget[/name] and [name]Katharine[/name]/[name]Catherine[/name] are the most intuitive. Though I think that any bird name would really work. Or you can just hope for a “bird”-like child. (I mean possessing traits that would usually remind of a bird not that she will actually have wings and a beak.)
I have a friend named [name]Katharine[/name] whose family calls her [name]Birdie[/name] (I kid you not) because she loved the book “[name]Catherine[/name] called Birdy” as a child.
[name]How[/name] do you feel about [name]Elizabeth[/name]? Since [name]Betty[/name] is a nickname for [name]Elizabeth[/name] and [name]Birdie[/name] is a nickname for [name]Betty[/name], it seems like a logical jump.
This might be a stretch, but what about something like [name]Larkin[/name]? or [name]Avery[/name]? Things that have a bird-like feel but aren’t literally related to the “bir” sound.
[name]Bernadette[/name], [name]Beatrice[/name], [name]Barbara[/name], [name]Brianna[/name]
[name]Perdita[/name] sounds a lot like [name]Birdie[/name]! [name]Perdita[/name] is a character in [name]Shakespeare[/name]'s [name]Winter[/name]'s Tale. Her name means “lost.” She’s a beautiful princess who’s abandoned and raised as a shepherdess, later to fall in love with a prince and live happily ever after.
Or how about just [name]Phoebe[/name]? They are such sweet birds, and it’s a fine, venerable name.
[name]Even[/name] Belphoebe could work, if you want something more elaborate.
I really liked the suggestions of [name]Beatrice[/name] and [name]Bernadette[/name], as both names were on my list anyway. I hadn’t thought of them, because it seems bit of a stretch to me. But my criteria is a little strict, and if I actually want to use [name]Birdie[/name], then I think one of these two will be my way to go.
Thank you, everyone! I appreciated all the input. I could tell a lot of you really took some time to help.
Lol, probably too late now but I was going to suggest [name]Bernadette[/name] too 
I would think the female version of [name]Robert[/name] would be [name]Roberta[/name] to constitute the nn [name]Birdie[/name]. I have a male neighbor who’s full name is [name]Robert[/name] [name]Bertram[/name] and they call him [name]Birdie[/name]. [name]Birdie[/name] is such a cute name for a little girl, I hope you use it!
Cute nickname! Kind of reminds me of Lady [name]Bird[/name] [name]Johnson[/name]
[name]Bernadette[/name], [name]Bernice[/name], or [name]Elizabeth[/name] (that’s a bit of a stretch, though)
Good luck!
On the tv show madmen [name]Don[/name] calls his wife [name]Betty[/name] (real name [name]Elizabeth[/name]), [name]Birdie[/name] as a nn.