I hate nicknames but Julip is kinda cute.
Also sounds like "tulip". Why not?
As for the nicknames, we've got JLip, Pogo, Radio, Tut, and Sot. We can deal with it.
I think it's better not to use nickname. This is bound to offend someone of different nationality, different language, different culture, or different religion.
I for one have never liked nickname for Plushenko, such as Plushy, plush, and plu. To me, those nicknames for Plushenko has always been connected with flashy, flush, and flu. I certainly don't like the sound of JLip or Julip, especially a formal request like this made it like asking for a permission to "officially" change the girl's beautiful name. JLip sounds to me like a name calling from a non-Julia fan though I don't know whether you are her fan or not.