I think it will be fine. My oldest daughter's name is Gwendolyn but I always intended to call her Gwen. I love long names for when you are older but Gwen is better for littles I think.
I also call the other 2 by names that aren't their legal names and no confusion
I actually went to school with a guy whose legal name is Clayton but he went by Mac. I went to school with him k thru 12 and didn't even know his name wasn't Mac until 3rd grade when the teacher said Clayton. They have a place on school forms for preferred name and they even announced his name as Mac during graduation.
Anyways all that to say go for it
Privacy? Who needs privacy