I'm not a huge
Lily fan but the names you have are good. There are so many
Lily Rose's about I don't think anyone will outside BtN will care about double nature names.
LLL initials are fine!
Double
Lily names:
Lily JuneLily ZaraLily MayLily TessLily FayLily JeanLily JoanLily ClaireLily JaneLily Belle (I knew one once!)
Lily JessLily GwenLily Maya