I'm not very sure as I haven't really happen to come across a card that let's you pay with crypto directly. Most people make use of exchanges or regular P2p transactions to convert from crypto to fiat to be able to make payment but spending crypto directly from a card isn't something I'm quite familiar with.
The same here, I have not seen any crypto transaction done with card, I believe such methods does not exist or is very rare because I wonder how someone would be able to connect their crypto card to their custodian wallet for payment of any transaction, however the only way I can think of is just the exchange transaction or the P2P you mentioned because since the OP is looking for a way that's more faster and less transaction fees those listed methods should be okay because it makes it very easier to make or initiate a transaction with someone through exchange without so much delays.