"So, I can't claim the full $4000"
Yes, you can. Since your student is declaring that much of his scholarship as taxable income, the tuition is no longer being paid by "tax free scholarship". It's being paid by his taxable income.
This is a well known (and frequently used) tax “loop hole”. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship. You cannot do this if the conditions of the grant restrict it to paying tuition.
This is not some sinister scheme. From the 2019 form 1040 instructions (pg 95): “You may be able to increase an education credit if the student chooses to include all or part of a Pell grant or certain other scholarships or fellowships in income. For more information, see Pub. 970, the instructions for Form 1040, line 18c, and IRS.gov/EdCredit. Page 16 of PUB 970 (2019) actually has examples of how to do the “loop hole”.
