Dina & Michael Lohan Custody Battle -- Lohans Drag Catholic Church Into War

Michael Lohan's gonna have to say lots of Hail Marys -- and fork over lots of cash -- if a judge orders him to ante up thousands for his kid's Catholic school education ... but Michael says it's just more Dina BS.

Dina Lohan filed new court docs -- obtained by TMZ -- claiming Michael refuses to pay for Lindsay's brother, Cody.

Dina grouses ... Michael hasn't paid a dime, even though they were supposed to split the $10K/year tuition. Dina says she covered the first 3 years -- $30K -- but wants Michael to ante up for Cody's senior year.

But Michael's says poppycock ... telling TMZ he already gave Dina 30 grand for the school and a car for Cody. Plus, he says he's given her $1,500 a month for the last two years. And he says he's given daughter Ali $35K over the past year for an NYC apartment.

Michael says what is especially galling ... Dina blew around $10K on trips to visit her boyfriend in Texas.