I think the main thing when you are negotiating at your meeting, is look at it from the companies POV - ask yourself what do they want / what problems can you solve for them? (People are terrible at not doing this.)
The main thing they care about is that you perform in the role and become a useful investment going forward. They’ve recruited you, and they don’t want to loose you. So sell it in those terms - give whatever you want a positive spin - it’s fine to mention the practicalities of working with kids, but don’t moan or sound like you can’t cope (you can do this as a known employee, not as a newbie).
Those are long hours outside the house - can you deffo not ask for a day WFH? Is there real evidence they won’t allow it? Unless there is hard evidence that’s the first thing I’d pitch for, explaining that even with your DH stepping up, you of course have a family to run, and a day from home would really help you keep your energy levels up.
If it’s not possible - then I’d ask for a 9 day fortnight - with either Friday afternoons or every other Friday off. In a year you can bring it down to 4 days if it’s working.
The other thing is trying to do the job and not having help with the house and family chores is really setting yourself up for trouble - get a cleaver ASAP. Get some family meals in from Cook or similar, so there are nights no one has to cook, do some batch cooking - and make your everything is divvied between you and DH - he sounds willing but I doubt he’s pitching in enough because neither of you is used to it. Get a shared calendar and share all tasks equally. And make sure the kids to the basics (with pocket money fines if they don’t.)
If you haven’t sat down w your DH to discuss this - do. You should be working together to solve it.
Also - if you are finding managing the work load tough, think about a few sessions with a career coach, they can be really good at getting you to see clearly how you can make things easier. A few sessions is usually fine.
Finally 8 months is not long to go from O to 60 so give yourself time to figure out how it can work for you. The money you spend off-loading work onto a cleaner etc will get paid back later as you have a well paid career - if this company isn’t family friendly enough for you - after 18 months you can look to switch.
Good luck