Divorced Parents Form 8332 And Months Child Lived With Me

I'm divorced, and the agreement is that my ex and I declare our child on alternate years. When I declare the dependent my ex signs form 8332 "Release/Revocation of Release of Claim to Exemption for Child by Custodial Parent", and I file it with my taxes

 

My ex works 3 days a week 12 hours each, which is considered full time. I take care of my child 3 days a week when she's working, so she can work. That is about 150-something days a year my child is with me

 

The problem I face is that even in the year that I can claim my child as a dependent, when TurboTax asks me for the number of months the child lived with me, I don't know what to enter... If I enter 6 months (b/c my days are less than half the year), I lose head of household, and I think I also lose some of the benefits of declaring a dependent. My refund decreases about $2K or $3K

 

So, even the year I can declare my child as a dependent, I don't get the full tax benefits of doing so

 

What would be the right way to handle this situation?

 

TIA

 

Nick