Honestly, OP you do come across as slightly hard work. It might be more honest to say you don't like SiL because she's an awful person who has treated you terribly so you don't really care if the date you have your baby affects her.
The list of random reasons why you "might" have this baby on the day her child died but with the implication that it's all out of your control are stupid. If you cared about upsetting her, other than the hospital refusing any other date - which is possible and everyone would need to accept, everything else is sortable one way or another.
You have 6 months, start looking at emergency child care now, find a local agency or sitters.com or something, meet a few people. Speak to the nursery staff and see if any do ad hoc childcare when not on shift or would be willing to. Speak to other parents, start building your village. Partner could possibly take emergency carers leave if needed.
Your Mum could drive you to / from the hospital and partner could follow in a taxi or on the bus, so would be available to carry bags and be there for your surgery. You could reasonably get a taxi there and back yourself with partner while Mum had the kids but I understand you don't fancy that, you'd be fine though, a car is a car and at least a taxi driver would help with bags.
A weekend is not long enough to recover from a section as you well know, especially if you have additional risks. Again, partner could book a taxi, 7 seat if needed for the school run to avoid the 50mph road and take all the children with him for a few days. It's an expense but save for this now, a couple of £ a week in a taxi fund jar. Or book an hour of childcare in the mornings for the first week if it works out cheaper so you can rest and concentrate on baby while partner does the school runs.
All of this is sensible to do SiL situation aside because you cannot guarantee getting a section between school hours on a Friday, you know birth is not straightforward and baby could decide they're coming early, you could only be offered say a Tuesday or end up pushed back by emergencies.
I'm sure there are other problems I've missed and that there will be holes in these suggestions but just take a step back from the emotion and think about it, you really do need a plan B.