Do I stay out longer unpaid or say goodbye to owning a home anytime soon ?

TL;DR it costs more than my rent for 9 days of childcare per month

Full story: my husband and I are both first responders in NYC, I work days (currently on state family leave, expiring soon), he works nights. Different rotating schedules, which I thought would be great because we would only need parttime childcare for 8 hours, 9-10 days per month. WELL upon speaking to a few recommendations and hearing the competitive rate, I might as well quit, I’ll be left with as much money in the bank. A CHEAP sitter comes out to $2k per month for 9 8hr days! A competitive rate comes closer to $2500. My husband makes enough to just cover our bills, I am on his insurance with the baby. I can take unpaid leave up to 4 years and potentially pick up random shifts at my side gig (I do also have 4 jobs I’m currently on leave from), but there’s no guaranteed availability, it’s random days when jobs pop up. If a 24 hr day care was an option I’d consider it, but I have not found one that exists as we need care from 3p-midnight. Grandma is willing to volunteer occasionally, but not all the days, and baby would have to travel to her, not stay at our place, which disrupts sleep schedules 😫

How do other shift workers do it? Any solutions or suggestions welcome!