The American Opportunity Tax Credit is a maximum of $2,500, not $4,000. There has to be $4,000 in QEE spending to get the full $2,500 credit. The credit is dollar-for-dollar for the first $2,000 spent on QEE, and then a 25% credit for the next $2,000 spent on QEE.
Edited to add: and I agree that using a credit card to pay college costs is usually a terrible way to pay.