Childcare for 6 month old on random days each month?

I'm not aware of any daycare with this arrangement. Your best bet might be using UrbanSitter to find last minute sitters but might be unlikely you'd have the same one every time if it's unpredictable. 

You're in a difficult spot because nannies and daycares are always going to prefer a regular, predictable schedule over random drop-in. The Bay Area is expensive and it's hard to make a living as a childcare provider.  There just are not many professional childcare providers who can afford to set aside "on call" time for occasional drop-ins.

Daycares rarely accept part-time children because they are limited by state licensing requirements (8 children for a small daycare, 14 for large daycares.) They cannot afford to reserve one of these spots for a child who is only there occasional days. Sometimes a daycare will accept two part-time children who come on different days, but it would be very difficult for them to find other part-time children whose schedules match your off days. 

A nanny share could be more flexible than a daycare, but again, you're at a disadvantage because more than 90% of nanny shares include just two children (see 2021 nanny survey) and most professional nannies need 35-40 hours a week.  If your child is only paying for a few days a month, the other family in the share must essentially pay the one-child rate on the days you're not there, so not a good deal for a family wanting a nanny share. If you knew which day of the week you needed childcare, you could find a family that has a nanny 4 days a week, and you could take the 5th day, but it doesn't sound like your job would allow that. You might try to find a nanny share with two full-time families who will accept a third child occasionally. But sharing a nanny with just one other family already is complex -- negotiating food, nap times, days off, holidays, and parenting philosophy. The nanny might be making a little more money when your child is there, but the other families are not paying significantly less, and they must do more coordination to add a third person. One way to do this might be to make an agreement with a friend who is in a nanny share who already knows you and doesn't mind the extra hassle.

I think what people do in your position is recruit a friend or grandparent or other family member who can babysit on short notice. Or try to negotiate a more predictable schedule with your employer. Or you could use one of the online services (care.com, urbansitter, etc.) to hire a one-time babysitter each time you need childcare. Or you could pay for full-time care even though you wouldn't be using it full time. Or, if you have an extra bedroom in your house, you could offer it to a student or au pair in exchange for occasional childcare, assuming their school schedule is flexible enough to accommodate your work days.

Sorry to sound bleak, but your job sounds like it is just not very compatible with the kinds of childcare that are available.

For that age, I really think it's important to have a consistent caregiver. It would be really hard for a six-month-old to have different back-up caregivers each time, so using a service like urbansitter or care.com is going to be a challenge. I think it would be worth it to find an arrangement with another family or families, and I think if you think creatively and can be flexible about other things (location, age difference between kids, etc.) I think you can find something. Example: a 3-family share where one of the families is flexible about which days they have care, so one family is full time and the two other families split the other half of the share and have to work out schedules to make sure the nanny always has two kids. We actually would have met this definition when we were looking for a nanny share because we had help from my mom that was flexible but she couldn't do full-time - I'm sure there are others in a similar situation. Or depending on how irregular your schedule is you could do a nanny share and guarantee a certain number of hours, and pay even on the days you don't need it - more expensive than only paying for the hours you need, but possibly cheaper than paying a solo nanny only when you need it. Another idea is to find a parent to swap with, maybe a stay-at-home parent or a parent with a flexible part-time work schedule - it's pretty hard to watch two six-month-olds but you could give it a shot. I would think about different possibilities like this and then make a few posts on BPN, nextdoor, etc., to see if you find any matches.