Yes, as per your example, there is $2500 of taxable scholarship income to report (difference between larger amount in Box 5 and smaller amount in Box 1) . In that case, the student would normally enter the 1098-T on her return to provide documentation for the taxable Scholarship amount (since she is required to file a return with a 16K W-2). Room and board are only expenses against 529 distributions reported on 1099-Q. She can also add her $250 in other education expenses.
You are correct that if you did not pay any education expenses 'out of pocket' you would not qualify for the AOC (which was designed to help parents with expenses they paid for education), and you would have taxable scholarship income added to your return if you added the 1098-T as issued on your return.
If you report as @Hal_Al suggests and add the taxable amount of Scholarship Income to the student's return while you claim 4K in Education Expenses, it's true that you may receive the $2000 AOC, but look at the bottom line in both returns to see what works best for you and how you feel about changing numbers on the 1098-T to report to the IRS to qualify for a credit.
If you decide to report the 1098-T as issued, since you wouldn't qualify for AOC (and only a $500 Dependent Credit), plus taxable scholarship income, you could consider letting your student file as independent and not report any education expenses. With a $12,950 Standard Deduction, her tax burden would be minimal.
Here's detailed info on Qualified Education Expenses. Here's an article on the American Opportunity Credit.
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