I'm going to take a different tack on this, and maybe I'm reading it wrong, but it looked to me like there was a situation of them agreeing to save up an extra amount (and sequester it in a separate account) to finance a SAH period, and then have her return to work when that money ran out. It looked like an agreement that they were going to put aside only so much money to replace her pay for this period of time, and that other monies were earmarked for other expenses. The distribution of who pays for what might be a holdover from when she was working, when those were the expense categories that she normally covered out of her pay.
I've seen families do this in the past, where the savings account essentially replaces the wife's income for that period. She draws from it as if it were a salary covering her contribution to the household expenses. Planning well in advance to do it this way can allow a limited period of SAHP when the husband just does not make enough to cover household expenses on his pay alone. Situations like these are not a case of *if* she will go back to work, but simply when. (And I know there are a lot of moms who had planned to reenter the workforce and found after birth that they didn't want to, but sometimes the financial realities are such that you have to get past that reluctance and stick to the schedule in the original plan.)
The wrench in the works in that case is that the SAHP fund money is running out faster than the OP had anticipated, and that returning to work sooner would be a financial problem because she had agreed to keep the stepkids over the summer. When she goes back to work their combined income will probably cover daycare for the baby, but not summer daycare for the older children as well.
If I'm reading this right, my first idea would be to take a p/t evening shift job if she can find one (leaving the baby with Dad), so that she can get some additional money into that account to get them past the summer custody period before going back to a regular full time job and starting to pay daycare.
FTR, DH and I do have separate accounts, and have had for the 20 years we've been married. We divide the responsibilities loosely in terms of who pays for what in regard to household expenses. However, we live WAY under our means, so unexpected expenses seldom cause a serious hardship for either of us; we just temporarily adjust who pays for what until the hole is filled. For instance, normally I pay for daycare, but last week I had a surprise $1000 dental bill that insurance refused to cover, so DH is adding day care to his side of the ledger for a couple of weeks. We can do that because we have ample wiggle room, but apparently the OP's family does not.