I also did this same exact thing yesterday. But we did have a temporary problem doing it. Day before yesterday, we were able to add Cheers to our cart, apply the AARP gift, see the balance reduced but then we were unable to use a credit card to pay the balance. Just wouldn't accept the CC. It is my belief this was just one of those buggy Carnival website issues.
Then yesterday we received an email saying we had an outstanding item in our cart. Went back to the website and were able to complete the entire transaction.
All this to say that you can pay for Cheers using an AARP for partial payment and a CC for the balance, but it's possible you might have to try it more than once.