Double-billed by Booking.com and the hotel. Customer service is completely lost. What is my next move?

I am stuck in an endless customer service loop and could really use some advice. On May 12, I booked a hotel through Booking.com and paid in full using my credit card. When I arrived to check in on May 17, the front desk told me they were unable to charge the virtual credit card provided by Booking.com and asked me to call them to get it sorted out.

I called Booking.com immediately. Their support agent told me they would talk to the hotel, resolve the issue as soon as possible, and assured me that I did not need to worry. However, when I went to check out six days later on May 23, the hotel informed me that Booking.com never actually fixed the payment issue. Because the hotel still had not been paid by Booking.com, I had to pay the hotel directly using my personal credit card. I essentially paid for the entire trip twice.

I have called Booking.com twice over the last few days. They keep saying they will look into it and refund me, but nothing ever happens. The front-line agents seem completely incompetent and act like they have never heard of a situation like this. I have to re-explain the entire story from scratch every single time I call. Has anyone successfully escalated an issue like this? Should I just bypass them and file a chargeback with my bank, and if so, do I dispute the original Booking.com charge or the direct hotel charge?