Lie-la is Lila to me, as I know one. I am not a fan of alternative y-spellings, so I would stick with Lila. Lilah looks like a shortening of Delilah to me.
Lila Elizabeth is fine, imo.
No, it doesn't make me think of Layla (how funny, I prefer that spelling, so maybe I'm not as anti y- spellings as I thought!) or Lola.
If the nickname is personal family use only, then you can use whatever you like. One of mine was known as Poppy for years, which bears no resemblance at all to her name