Is it possible to pay a hotel without a physical card (booked via booking.com)

I booked a hotel through booking.com – where you pay at the property.

However, my credit card provider – the Sainsbury Bank – has massively dropped me because of the slowness with which I send my card. I was approved and the map did not come. I ordered a new one and the next day the card arrived (almost 4 weeks later despite a delay of three to five business days)

Whatever it is, it seems that the new credit card is not coming. However, I activated it online because the card number has not changed, only the month.

I've since bought things online with this card (ie one that has not arrived yet)

Is it possible to agree with the hotel to simply bill me via booking.com, since I may not have the physical map on time?

Thank you