Whilst childcare seems to be the problem here @FriskyHeeler... I thinks it's a lot deeper than that.
The real problem is that you can't WFH 2 afternoons a week, because your team are incapable. You've made some comments about you being indispensable, your team being clueless, your line manager not picking up slack while you are off, how nobody covers and you just always pick up the pieces, how staff turnover is very high sknce you starte dthere. I would say that is terrible management on your behalf.
If you can't train your staff properly and rely on them... you don't get to reap the rewards of flexible working unfortunately.
Also with 3 young children, 2 of whom are SEN, one who is only 3 so not self sufficient.... WFH isn't a replacement for childcare. I WFH, and my youngest goes to childminder unless his Dad is off work. We do have a teen also but he's self sufficient so doesn't need looking after. You cannot possibly adequately work and look after 3 children. It's physically not possible to get the same standard of work done.
I have staff below me who I have allowed WFH 2 days a week, to facilitate drop-off and pick-up for creche / school... but they cannot look after their children and work simultaneously.
I think the beat solution here is perhaps a) upskilling your staff... you need a number 2. Someone who can immediately slot in if you called in sick, wgile you are on leave, someone who can do the day to day while you take up the running on the new client project and focus on the new incoming staff. Perhaps you could identify someone internally or ask to be involved in the recruitment process?
Secondly I would ask for flexible / compressed working week. Can you cut your hours by 2 1/2 days a week? Can you work an extra 2 hours 3 days a week while your DH is on his WFH days? Could you take your lunch hour 2 days a week to pickup the kids and bring to your mums house to look after them until either you or DH finish work for the day?
You can't just go in to a meeting with work, who are clearly seeing some issues or concerns, and basically day I'm so indispensable and I don't have childcare (for whatever reason, some of your reasons don't seem legit I won't lie but your choice) so you won't fire me and I'll just continue on even though this is clearly a shit show.
If you can improve the team... there's more chance of reaping the benefits of flexible working.