I wouldn't. I'd put whatever longer form you prefer on the BC. My name is a shorter form (very dated now but popular at the time) of a longer name, and I've always hated that I "can't" lengthen it, whereas if I'd been given the longer name I could have chosen to shorten it. My parents thought the short name was nicer, but the longer name would have stood the test of time much better.
I have a Frederick. He gets the full name from my DM, who loves it, Freddie from DH, Fred fom everyone else. He also has a nickname which is generally used by the family interchangeably with his actual name. To me, Fred is not a diminutive for Wilfred (that'd be Wilf) or Alfred (that'd be Alf or Alfie). I think people will assume your Fred is a Frederick.