Frankly, that's going to be up to the auditor if you are audited and I think you are not on strong ground.

For kindergarten and above, tuition is not eligible for the credit except to the extent that you pay extra to extend coverage hours so you can work (and the cost is itemized separately on the bill).  

So let's say public school kindergarten is 9am to noon and there is a program that allows you to pay for care from 8-9 and from noon-5.  That cost would be eligible if billed separately.

What you want to say is, I am using a private kindergarten that is 8-5 because my public school is only a half day, therefore I should be able to claim the equivalent cost of 8-9 and noon-5 care for the credit.  But the tax code is not written that way.  If private kindergarten was 9-5 and you paid extra for 8-9 and 5-6, that extra would be covered (if it was billed separately).  And if you went to public kindergarten and then paid a sitter for coverage, it would be eligible.  Even though you can argue that what you want to do is the functional equivalent, the tax code isn't written that way and I think that if you were audited you would lose.