Dependent Care Credit - Claim full tuition year?

No you would not enter the full amount. Preschool Tuition can count toward payments made for the Child and Dependent Care Credit, but you may only count the expenses in the year the care is received. 

They will not be deductible in the later year unless the expenses were actually paid in that year. See IRS 2021 Publication 503 Child and Dependent Care Expenses, page 10 & 11.

• Expenses prepaid in an earlier year. If you pay for services before they are provided, you can count the prepaid expenses only in the year the care is received. Claim the expenses for the later year as if they were actually paid in that later year.

When you prepay for child care, you claim the expenses in the year that the care is provided. So you have to divide the total based on the amount of time. The amount that was for 2021 you claim on your 2021 tax return. The amount that you paid for 2022 you will claim on your 2022 tax return, even though you paid it in 2021. This is an exception to the general rule that you claim expenses in the year that you pay them.