@3mealsaday
You need to tag team and he needs to do the weekends if you're taking time off to care for DC during the week.
So rather than late nights and early mornings, you need to be able to walk out of the house on Saturday and Sunday to go somewhere quiet (cafe, relative's house, lock yourself in upstairs bedroom) and work for an uninterrupted 8 hours while he cares for the DC. Not an hour here and there.
And if that's not enough, you need to do shifts. You're on while he's at school, he comes back at 3.30/4 and takes over until bedtime and then you both finish off any other work you have after bedtime (and share night wakings).
This ^
I think best to be very firm with your DH that if you're looking after children during the day, then he needs to come home as soon as classes finish, cook dinner etc so you can catch up later
and this ^ says similar. As teachers, we are committed whilst the dc are in school, but do have as much flexibility as the wfh person for all the other hours.
Your "flexibility" would involve you possibly looking unprofessional at work and working until some stupid time of night - you mention midnight.
Well, it might surprise you to know that that is how work and having small dc pans out sometimes. Lovely for those that haven't had to do it, but both dh and I have had various spells when our dc were little when that is what we did. Is it ideal ? No. Is it doable for a short period in your lives? Yes. Both parents working and having 2 small dc is hard. I don't think anyone is denying it. What you have to do, as a couple is do whatever you can to get through it. This is the worst time. Work together to survive.
I was teaching in school when my dc were little and now my job is mostly wfh where a lot of the time I manage my own diary and workload. I 100% agree that the person wfh has to pick up more of the looking after poorly dc during term time (keep in mind that is already down to 3/4 of the year), as they literally can then work on a Saturday or Sunday (or both) as well as evenings to replace those hours, but a teacher can't say to his class - "I'm not here for maths, Period 4 on Thursday so can you all come into school on Saturday at 11.30 to replace the lesson I have had to miss". Like the pp who said their dp was a clinician - it is just the same. Some work can be still done perfectly well at 8pm on a Wed night or 2pm on a Sunday, and other work can't.
OP's dh hasn't said they won't ever take the time, they have just suggested that the partner that works from home and manages heir diary is better placed to do more of it. Which is right.
Anyway OP, hopefully this is more of a worry than it will be reality. Although it seems so at the time, dc are not generally poorly that often. Now they've built up their immune systems a little bit, it just might not be much of an issue over the next 6 months and everything will settle down.