It's possible that Luna has moments in which she feels bothered by Lily. Try to give her quality time as often as she wants it.
Our boys have developped a totally different time schedule for love. E.g. Filou comes first thing in the morning, says hello, follows me to the bathroom and gives headbumpies. When I go down, Filou stays upstairs and now it's Tigris' time who lies on the newspaper asking to be petted.

Maybe you can flirt with Luna while Lily is sleeping. Anyway in addition to antibiotics much attention will be good for her
(Which does not mean she doesn't love Lily -I am sure F+T love each other too).