I had a similar issue with an Asian airline, except my problem was that I lost the credit card used to book the flight earlier that day.... but their anti-fraud procedures required I show the card.

The airport counter offered the same solution. So I "bought" a 2nd ticket right at the counter with a different CC, and once I got back - I had to contact the airline. It took ~2 week and I had to fill out a form indemnifying the airline in case I somehow disputed a charge for a ticket in my name, attached to my passport (which they scanned).

In the end, the 2nd ticket was fully refunded. But I was required to carry the $2k+ USD charge on my card for ~3 weeks total.