Sounds like DH could easily do 6 of the drop off/pick ups on his WFH days, so that needs to be the new default - even before your new role starts, so both he & kids can get used to it.
He's probably right, he will find it stressful because it's an organisational task he's got away scot free on until now, but it doesn't require ovaries to perform it, just practice, so he might as well start now!
Understand that his boss has already been clear they won't offer compressed hours - that's more common & not unreasonable. If a role has work coming in evenly over 5 days & specific turnaround times, it's not fair on the team if one person isn't in 20% of the week. Depends entirely on job and is a genuine 'needs of the role' reason for refusal in many cases.
But it seems much less likely a request for flex hours to either start at 9.15 or leave at 4.45 on his 2 office days would be refused- it might be unusual for his boss, but DH should at least ask!
Is there an option for him to take Friday as annual leave on a Friday for 3 of the 6 months. Obviously it wouldn't leave enough for a family holiday together this year - but that might be part of the 'doing everything he can' to help you establish in your new role & get settled in. Maybe Friday in childcare over summer holidays while there are more options available for 6 of the weeks?
You've just secured a new job, presumably for £X amount more salary.
If his choices are 'the boss will look poorly on me, but approve the request' or 'lets just fork out £4k a year for Friday childcare', I think the first option is head & shoulders above the second one! Unless there's an impending promotion for his worth £4k post tax in the bank on the immediate horizon?
Mum's wheedle and flog themselves to fit hours around other commitments. Even my DH's dinosaur boss adjusted to the diary block that DH WOULD be leaving the office at 4.50 2 days a week so the children weren't abandoned on the pavement outside club. These bosses who never had to lift a finger themselves for childcare CAN adapt to minor working hour tweaks at a push!
As a final warning though...if DC starts school in September, keep in mind you might BOTH need to have some flex/annual leave in hand until October half term, as lots of schools bugger around with mornings one week, afternoons another/finish before lunch/finish after lunch & who knows what other mucking about until kids are actually in full normal school hours, as part of 'settling in'. 🤞 for you all they are not one that strings it out for too long! But even my DC who had done 4 10 hour days in nursery for yonks found a 7 hour day in school harder, because they were being told EXACTLY what to do & where/when they were allowed to play or not in a different way from nursery. They really liked 'playing teacher' at home in that first term....mostly dishing out instructions & telling people where to sit/what would happen next 🤣 It's a big adjustment for them after nursery, which is mostly free-flow play.
Congratulations on the new job @Stuckbetweenarockandarock, you're doing really well & if Friday in childcare for 6 months is what keeps your back from breaking after all, don't beat yourself up about it, you're doing great! 