I'm no expert on this subject, but I've got a couple of suggestions.
1. Start tracking the clicks - in the UK it would be a bit of a grey area if you recorded any personal data without informing the user, but you can certainly log what was clicked and when. Later, you can use that to present your business case to the credit card companies.
2. Contact the credit card companies now anyway. Ask them if they'd be interested in some kind of arrangement. They may have a business model in place to deal with this - either they pay commission or they would pay to advertise. You could also ask them to provide an electronic feed of their latest offers.