Can I reverse a payment and re-pay with a different debit/credit card?

Yes! You can certainly reverse that payment by requesting them. It's possible as I had a similar experience a few months back and the owner of the store obliged my request in a very professional manner.

It's a bit of an imposition. Not only will the store have to do the work of processing the additional transactions, but depending on their payment processor, they likely will incur additional fees. If it would be a significant problem for you to have the payment on the original card, go ahead and ask, but if it's just a mild preference (for instance, because you would have gotten 1% more rewards on a different card), I wouldn't do it. Ultimately, it is up to the store, and they have the option of refusing, but it's still taking advantage of their generosity a bit.

Regarding your "chargeback" tag, you DEFINITELY should not accomplish this through a chargeback. The word "refund" refers to a return of your money processed through the merchant. "Chargeback" refers to a return of money processed through the credit card network's fraud policies. Doing a chargeback means hefty fees for the merchant (even if it's not upheld), and harms their relationship with the other parties (their bank, the credit card network, your bank). Furthermore, doing a chargeback without valid reason is fraud.