I would recommend splitting the costs 50/50. IRS will allow the full child care credit based on the maximum expense of $6000 even if the prorated payments are not 50/50.
The Child and Dependent Care Credit can get you 20% to 35% of up to $3,000 of child care and similar costs for a child under 13, an incapacitated spouse or parent, or another dependent so that you can work (and up to $6,000 of expenses for two or more dependents)