$4000 is the amount of expenses you need to claim the maximum $2500 amount of the American Opportunity Credit (AOC). So, it would be $2500 that you might lose. But, you don't have to lose it. Claim $4000 of expenses on your return, for the AOC, and have her claim $4000 more as scholarship income on her return.
Furthermore, even though the $16,000 spent in Spain was not on the 1098-T, some of it is still a legitimate expense to use tax free scholarship for. The part of the $16,000 that went for tuition and fees is tax free. The part that paid, for travel, food and lodging is not.
