If you can recieve payment from Paypal, have you tried using your balance? Paypal doesn't automatically transfer your money to your bank account/cards. It'll just keep the money you have there until you're ready to transfer it or use it for online purchases. I've done that in the past, and it's worked out fine. No card needed.
I would also look a bit more into your card issue. Which bank are you using, and do you know if they are accepted by Paypal (I would assume more bank cards are accepted, but I dunno)? Has this happened before with other online purchases (such as using Amazon)?