I would just logon via smartphone and show anyone who questioned. I've never been in your exact situation, but the only place I needed to show a physical card to someone was at the Alaska Boardroom when trying to get in as a Sky Club member.

Out of curiosity, how does the relationship with Virgin Atlantic work? I know that for the SkyTeam partners, the big hurdle is hitting Gold status as that gets you SkyTeam Elite Plus and the benefits beyond that are negligible (maybe op-ups?). Is there a big difference on Virgin Atlantic between Delta Gold and Platinum? Do you have your GM card just in case?