Just name her Livia and have her friends call her Liv. No need to give her the name Diana too if you aren't even going to have her go by that name.
It's silly enough in real life when a person only goes by their middle name or a nickname for the middle name. But at least in real life they can say "Well, there's so many Michelles wherever I go that it's easier if people just call me Kathleen." Or "My first name is Theobald. My middle name is Robert. What would you do if you were me?"
In a story, the character really can't go into all that, and especially the writer can't if it's third-person, without it seeming totally unnecessary.