Kind of. She will be doing a lot of explaining of her nn. But with that said, I grew up with a girl we all called Annie. Her name was actually Ann Elizabeth. Her mom had originally spelled her nn Ann E., but she changed it when she got in school.
I don't think she will be doing a lot of explaining at all. As someone who has gone by a NN my entire life, I am rarely asked what it is short for. But if she is, it is a quick explanation. As someone already mentioned, it is a legit NN for Elizabeth if she didn't want to go into the initials. That being said, I dated a guy who's name was Les. It was a nickname, it was actually his initials (his full name was Leonard Earl Stokes). It was never an issue for him to explain that, and no one thought it was weird.
Either way, I think it is a really clever solution and I like Ellie a lot more than either Lauren or Elizabeth.