I called my DD Lula as a NN (real name Lucia) but I'd never ever use it as a full name- sorry I don't like being negative. How about Lucie, Lucy, Luella, Talullah, Louisa?
I think that if she chooses to have a sensible career she may appreciate a more formal name, you don't know what kind of person she will be.
The other thing to consider with Lulu is potential playground taunts- 'loo'? etc. I don't think the botoxed scottish singer will feature that prominently on the radar of children her age.
Alternatively if you really want that to be her name (which I understand, tbh, if that's her name then that's her name and everyone should call her it) then choose a 'sensible' middle name, so she can pick and choose.