You have one main issue (can you claim $4000 of expenses for the AOTC) and numerous insignificant side issues.
Q. Can you claim $4000 of expenses for the AOTC?
A. Yes, your math is correct. @KrisD15 has explained the simple (and correct) way to do that. "Enter the 'mock' 1098-T with the 4,000 in Box 1 into your TurboTax program. Finish the Education interview as if the 4,000 was the only expense and there were no scholarships and/or financial aid.
Go ahead and file you tax return without regard to any of the side issues. They don't affect your filing.
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Side issues:
1. You don't need a 1098-Tto file. But, it would be nice to have it for your files. Since the school is willing to give you one (that's unusual), go ahead and get one.
2. Your daughter's amended return. As explained at your other post*, an amended return is technically not required because her total income ($6125 wages + $5326 taxable scholarship) is less than the filing requirement ($12,550). You can file, and claim the AOTC, whether she amends or not.
3. You may totally disregard the 1099-Q since there is sufficient room and board to cover the distribution (although the "$2,800 of that being paid directly to the student" is a little iffy). But that doesn't affect your return.
4. "Does it matter that I haven't located any info stating the scholarships must be used for tuition"? No. just the opposite. Unless you know otherwise, you may assume the scholarship doe not have to be allocated to qualified expenses.
5. Is it a problem that the child already filed and did not include this income? No and yes. No, you can still file your return without it. Yes, it would be nice if her return agreed with what you are claiming, but it is not necessary if your records are good (in the unlikely event you/she are audited)
6. I suppose the parent should wait to file after the child amends? No, it's not necessary. She can amend anytime. IRS matching doesn't come until much later, and probably not at all.
7. Who enters the book expense? Or is nobody entering the $1,078 for books? Nobody, because you're both using a workaround (just entering $4000 total qualifying expense in your case and just entering the taxable amount in her case).
8. Should the student enter the taxable scholarship in the educational expense/scholarships section of TT or the "Less Common Income" section? Both get the correct result on the student return. But I think it is better if taxable scholarship is entered in the education section, rather than the other income section, as that gets TT to place the SCH5326 notation on the dotted line next to line 1 of the 1040.
*https://ttlc.intuit.com/community/college-education/discussion/tuition-tax-credit-with-a-scholarship...
