As to the $1584 scholarship, you decide which year it goes against. Just be consistent . If you treat as 2022 on your 2022 return, don't treat it as 2023 on your 2023 return. I would call it 2022, based on your description. This is a common problem, we see in this forum.
Based on the results, it appears you you have successfully enter things in TT (I get $1782 also*). There should be no penalty (having scholarship and using the AOC are penalty exceptions). Verify that on form 5329.
There is an alternative: have her claim the $3834 of scholarship as taxable income**. That reduces the $1782 to about $204. Saves about $140 in tax. It may get a little tricky entering in TT, but I think I can walk you through it.
*3404 + 7262 + 6139 = 16805 QEE (qualified educational expenses)
16805 - 4000 (used for AOC) -3834 Scholarship = 8971 QEE available for 1099-Q
8971 / 13300 = 67.45% tax free
0.3255 x 5472 = $1782 taxable portion of the distribution
** So, how can reporting $3834 of income be better than reporting $1782 of income? Scholarships are a hybrid between earned and unearned income. It is earned income for purposes of the $12,950 filing requirement and the dependent standard deduction calculation (earned income + $400). The $1782 is unearned income. So, even though it's reported as income, it doesn't get taxed. It is not earned income for the kiddie tax and IRA contributions.
