In this sort of situation, a double backer is your
best friend. I always leave mine in the card box, or place it in my wallet. In a cinch, this allows you to get out of most mess-ups, using a fun method I thought of:
Simply ask them what their card was as if you didn't know, and spread through the cards pretending not to see it (actually cull/cut it to the top

) From there, however you want to present it, go and get the double-backer from the box (or wherever you put it), place it on top of the deck, and execute a double turnover. Because it's a double backer, you can cleanly push off the top card (their card), and be clean without making it apparent that you messed up at all.
That is my go-to method for whenever I mess up, and I'll often even perform it as a standalone effect because it's great as a regular trick as well as a copout.
p. s. If you're worried about the leftover double backer, just cut it to the bottom and cop it off, not difficult while they're reacting to the trick