I call my wife that occasionally, but only when I ask what the word is ("what's the word, little bird?")

That was the nickname they used for Tevye’s daughter Chavala in Fiddler on the Roof.

If I ever refer to anyone as little bird it'll include miniguns.

https://en.wikipedia.org/wiki/MD_Helicopters_MH-6_Little_Bird

it should be something that comes about organically, not something forced.

Ok, but my pet name for you is now Hemorrhoids Henry

Well my significant other is not creative at all, and started calling me poop-y, which i dislike (there are so many words, why pick a term for excrement), In that case, I had to speak my mind and explain that I did not like that pet name. Sometimes, nature needs a little help

One day my wife said "see you later, alligator" and I reflexively replied "bye gator" and she's been gator ever since.

My wife has thousands of cute nicknames for our dogs. She calls me by my first name.

Aww, Pencil Dumpling... You're not missing out on much!

I love you, sour meat 💖

Sometimes it's exactly what you want, though.