If the priest is married, his wife can care for the child.
This dilemma doesn't seem to be a problem for the married doctors I know.
We have married priests in the Church today: Anglican converts, three of whom are pastors of very large parishes in Ft. Worth.
That's far from "part-time."
I used to be against the idea of married priests but the Eastern Orthodox seem to do okay with them. I believe that the way they do it is they allow married men to become priests but priests don't marry ---so you wouldn't have your parish priest dating.