So do it without the log. There's no log of alcohol and tobacco buyers, and also no need for "audits" there. Have some undercover inspectors do spot checks to make sure all retailers are checking ID. Fail the test, lose your license.

“we have no record of this user gaining access”

I don't understand. Who's "we"?

The store sells a scratch card containing a single-use code, after you show them ID. When you sign up with whatever social network, you enter the code and the social network validates it with the code provider service. That's it.

At best the code provider can tell the social network which store the code was purchased in. Drive over to the next state to purchase codes if you're paranoid. Mandate by law that no records of code purchases or copies of ID are kept. We don't require it for alcohol or tobacco purchases so there's no justification here.