AFAIK you, or the SIC that is, needs to have C680 SIC only in the FAA license. I'm sorry that I can't point you to the "proof" but that is the way it is and that is the way I used to fly (N-reg a/c since 2004 in my case).
When you do the 61.58 just ask the examiner to put a SIC only on your license.