Here is the list of CSS Profile schools, you can sort by which ones require NCP financials. Verify this info on each school’s site.
https://profile.collegeboard.org/profile/ppi/participatingInstitutions.aspx
You pay to send CSS Profile to each school, so you control the info that is sent. For U Chicago, you can complete their own form, rather than pay for CSS Profile. Basically, you have to go to every school’s FA page to see the requirements. Applying for Aid | College Admissions
Both Vandy and Chicago reserve the right to collect NCP financial info.
Note this year the definition of custodial parent for fafsa (and therefor CSS) is the one who provides the most financial support, not the one the student lives with the most. That doesn’t sound like it would make a difference for you, but in case it does.
Have you run the net price calculators of the schools on your list? You can run them with and without bio-dad’s financial info to see the difference.