Because you have a cousin named Lila I wouldn't consider Lilac for a first name. I feel like Lilac, which is one of my favorite girl's names as well, would be mistaken for Lila quite a bit in general. Not to mention if she has the same nickname as your cousin's name!
I can't think of any nicknames for Lilac other than Lila or Lil, which is not very intuitive.
It bums me out! I could always use it as a mn
Maybe Lyle (though it's really masculine for Lilac). How often do you see your cousin? That could make a difference in whether its too close. There are quite a few Sams (male and female) on one side of my family and that doesn't seem to stop anyone from using it. However, that side of the family is quite scattered about and only gets together on holidays.
For me it would depend on how close you were with your cousin. Personally it wouldn't deter me from using the name but if it would cause family issues than I would avoid it.