Since you filled out the eDV registration with a 'married' marital status, you should use the date of your proxy marriage on your DS-260 form. Fortunately for you, the US immigration law recognizes proxy marriages - the tricky part in your case of course is the fact that there's no official marriage certificate for the proxy marriage and your sworn affidavit also clearly states your marriage wasn't officially recorded. But I'm assuming some sort of ceremony did take place for the proxy marriage - do you have evidence of those such as pictures showing both parties parents, other family members, friends, documented plans, expense receipts, etc? You may need to present those at the time of your interview to back up the proxy marriage claim since there's no marriage certificate to support it.
And when you do send in your AOS package, do include copies of both the affidavit and the US marriage certificate as further proof you guys are actually married. Having on hand, at the time of the interview, evidence of joint assets, bills, pictures of both of your before and after the marriage, joint activities, etc, could also become useful at the time of your interview.