Onboard, they apply the discount at the point of sale, though. They charge your card at the conclusion of the cruise.
I suppose if you totally switched your payment method, they could try to go back and reverse the discounts, but that sounds like a nightmare logistically. I am betting that if you paid down the balance with another card and then let a few bucks charge to the Disney Visa (which was initially registered as your main form of payment), it would not matter.
Honestly, not that much stuff is discounted for the Disney Visa , anyway to really worry about it unless you are spending thousands on merchandise.
The other benefit to the Disney Visa is the 6 months free financing