You can pay private school tuition from a 529 education account. That may provide some tax savings.
https://www.irs.gov/forms-pubs/about-publication-970
Regarding the dependent care credit, the credit is available when you pay for care for your dependent so you (and your spouse if married) can work full time. Expenses for school tuition at Kindergarten or higher are not considered "day care" expenses because your child must be at school anyway—you are not paying for care so that you can work. However, if you pay extra for before- or after-school care, that is eligible for the credit, provided the cost is itemized separately on your bill. (For example, if school ends at 3pm and you pay extra so you can pick your child up at 5:30, that extra is allowable.)
Day summer camp is eligible because you are paying for someone to take care of your child while you work. However, overnight summer camp is not eligible.
"You can include the cost of a day camp, even if it specializes in a particular activity, such as computers or soccer. But don't include any expenses for sending your child to an overnight camp, a summer school, or a tutoring program."
https://www.irs.gov/forms-pubs/about-publication-503
