Simple answer: You cannot claim the full $2500 American Opportunity Credit (AOC). You cannot use the full loophole (see description below) because her school scholarship is restricted to being used for tuition ("school scholarship awarded to her that could only be used to tuition at the school"). That is, none of the $6331 in box 1 of the 1098-T can be claimed for the AOC.
However you can (probably*) count the $1900 in books, fees, and equipment. The first $2000, of qualified expenses is a 100% credit, so your AOC will be $1900.
*See Q5, Q6, and Q7 at https://www.irs.gov/newsroom/american-opportunity-tax-credit-questions-and-answers
She should already be reporting $3139 as taxable scholarship (11,370 - 6331 -1900 = 3139). For you to claim the AOC, she should report $5039 (11,370 - 6331 =5039) as taxable.
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You essentially have to use a work around in TurboTax (TT). Here's how I would do it. Enter the 1098-T, on your return, but only enter $1900 in box 1. No other numbers. You only enter the 1098-T to get TurboTax to check the proper box on form 8863. Lying to TurboTax to get it to do what you want does not constitute lying to the IRS. What you input is not sent to the IRS.
Enter the 1098-T, exactly as received, on the student's return. Enter book expenses separately. In his/her interview, you should eventually reach a screen called "Amount used to calculate education deduction or credit" Be sure the amount in that box is $1900 (TT will not automatically populate it with $1900). That will put all his excess scholarship as income on his return.
Be advised some people are saying they're not getting the "Amount used to claim the tuition deduction or credit" screen on the dependent’s . The alternate workaround is to enter $1900 less than the actual box 1 amount, when you enter the 1098-T
There's yet another (and simplest) work around. Manually calculate the taxable amount of scholarship ($5039) and enter the 1098-T, on his return, with 0 in box 1 and the taxable amount ($5039) in box 5. In that case be sure the amount in the "Amount used to claim the tuition deduction or credit", if it show up, box is 0.
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There is a tax “loop hole” available. 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 school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.
Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.
Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket, she would only need to report $5000 of taxable scholarship income, instead of $6000.
