I agree, I think it would be too much to just switch the name right now, especially just because you think it’s too popular.
How about a nn that isn’t necessarily related to her name, like [name_f]Honey[/name_f] or [name_f]Ellie[/name_f] (Lily starts with L)? You could even do something based off of your last name, and then she could choose to be [name_f]Lily[/name_f] or keep that name when she gets a little older!
I know it’s frustrating, but as someone who grew up with a very common name and ended up choosing to go by my middle when I got older, you really can’t control how these things are going to go unfortunately.
Best of luck to you and little Lily!! (I think the name is gorgeous by the way) 