Can someone help me with the name of this children's book?

I read this when I was younger and loved it. I don’t remember much but it was about a little girl who was given a stuffed rabbit or she found it, not sure. I want to say the girl’s name was Elizabeth but can’t be certain.

I do remember she performed pretend surgery on it when it was missing an eye or stuffing. Then the stuffed rabbit watched from the stairs as the girl had a birthday party. Not sure if the rabbit felt left out and neglected but that’s what I think happened in that scene.

I think it talked about the girl growing up and not playing with the rabbit as much.
I don’t really remember the ending.