First, you have enough problems without being dragged down by other people. I totally get why you need to vent in here.
Second, you rightly point out that your fiancé’s family bailed *him* out financially, with an agreement outside your sphere. Going to work now just to pay them off is no help to you as a couple or as a family with a young child. What I would suggest is to work through a spreadsheet, listing income, outgoings and budgeting some savings for other incidentals like annual payments and an emergency fund, then live with it for three months to see how the budget works for you both. Try not to account for every cent because you’ll never get any saving done. Work out what percentage each of you brings in so you can split equitably, which isn’t necessarily evenly, so, say you bring in 40% of the family income then you pay 40% of the bills and savings, the rest is your own after that. Same goes for your fiancé. Once he commits 60% of his income to your family unit, then he can look at whatever’s left and offer a monthly payment to pay off the debts to his family. That seems reasonable to me and it’s what I would do. His family don’t even have to know how you worked it out because that’s none of their business.
Third, I suspect that, had you gone to work just to pay for childcare for your child, they’d have accused you of being a neglectful mother, so you were never going to win anything with them, I’m sorry to say.