To establish if a child minder can be flexible enough for you will simply take a phone call. This is a tough puzzle! (It feels like a maths coursework problem to solve)
Sadly I think keeping the jobs with the moving shifts will likely cost you. Yes some providers will be flexible, the majority (and I do believe the majority having spent a decade working in early years child care) will not. My old nursery was asked this frequently.
Nurseries are very very unlikely to allow such flexibility, however you will likely find one that matches the actual shift hours. The trouble is, yes you only need two or three weekdays most weeks, but to move it around means they have to hold a space for your child everyday, which will stop another child attending. So the only way most nurseries will work for you is to bite the bullet and pay for 5 solid days. But that will be very costly obviously. The hours however could work, around here most nurseries open at 7, and close at 6. (Just be warned though, opening at 7 doesn’t mean you can drop child off at 6:45 to make it to your 7am start - not saying you would, just something to consider. Certainly at that time of the morning nurseries will be strict and won’t let you in till 7 on the dot. This is my experience and I was like this myself!). You’re also unlikely to find a nursery open at weekends - unless you happen to live near one of those super flexible 24 hour nurseries (the only one I know of is in London! But they do exist - somewhere in the depths.....)
Is there likely to be a day in the week which you will both always be working on? If so it could be worth booking that day at a nursery and enquiring as to whether they have space for overtime. So for example you child always attends on Monday, but you then book extra sessions as needed. We never turned down extras if we were staffed, it’s extra money. But not a guarantee - there’s a lot of weighing up to do here.
Child minders would be the best place to start I think. There’s no harm asking. There may well be some lovely child minders willing to work with you on this. I would begin this way.
The nanny would be the ultimate answer - but another expensive one. Many nannies would be happy to do nights, weekends etc (because most people who become nannies know it’s a different type of role in general). However if you need them to be available at different times each week you will have to pay them a full time salary, because even though say you only need them two or three times a week, they can’t get another job because the days for you keep changing. (Take it as someone who once had a family at my nursery, who wanted a nanny, but only wanted to pay them the actual hours they were needed, yet also wanted them on standby 24/7 in case of work hour changes. They didn’t understand why no one applied)
I feel for you OP. Ignore people who say things like “why did you have children when you have no plan” and such - it’s mean and unnecessary. However do be ready to either make a temporary career or financial sacrifice for this. Childcare isn’t cheap (especially because the payment for babies is often higher to make up for the funded hours for the older ones)