They each have equal capacity to earn fairly generous incomes. They agreed that each would cover the expense of the child during their half of the year.
She has decided to take off work during her custody period, and has asked him to pay her a stipend during that time. Is she willing to do the same for him, if he decides to take off work when the child is staying with him?
That would be my verdict. Instead of alternating weeks, let them alternate months. Let him pay her the $4k during her month off, then when he has custody and she's back at work, let her pay him.
She left a job paying 240 a year to stay home and he bitches about 4 k in support? That is peculiar.
But the fair thing is for both to have 1/2 time jobs,,they could job share since she can't get 1/2 time. And then both contribute and both get to stay home.