AIBU to expect DH to do screen-free mornings with toddler?

This is a case of pick your battles. It’s fine for your toddler to watch Sesame Street, you will need some flexibility and don’t try to hard to be perfect parents- it’s impossible and leads to stress and resentment.
However, it does sound like your dh is not as supportive or understanding as he could/should be. Disbelieving that it’s tiring to be pregnant is a bit of an alarm bell. I would be concerned about the communication and not hearing each other, especially as it will be harder when dc2 arrives.
If you complain about him letting dc watch telly, when he thinks that’s fine your having a ‘battle’ about something inconsequential bit this leads to defensiveNess or dismissivness about the more important things.
It would be good if he was more supportive but if he feels that your criticising his parenting choices then that often leads to the man pulling back and doing less to avoid being told off.
Have conversations about parenting plans, ideals, etc as specific constructive chats, not in the moment when emotions can run high and people feel defensive or annoyed.
You want him to do thinks differently, but you can only adjust or change the things you do. But there is a lot of agency in this, approaching things differently will invite a different response from him. If he continues with things like not believing that it’s tiring to grow a baby 🤯, not listening to you, it might be helpful to have some couples therapy. Or read a couples self help book as therapy isn’t feasible with a new baby on the way.

Sounds like my horrible ex who shouted at me and laughed at me and told I wasn’t living in a refugee camp when I told him I was at breaking point 8 months pregnant

the what to expect when you’re expecting book has little grey paragraphs for dads to read quickly throughout the book but if he doesn’t care about your wellbeing he won’t read them just like my ex didn’t

he is really lazy

I would not be playing and doing sticker books at 5am. Thank god for TV and screens. You need to chill out a bit. My dc are 14 and 22 now and are perfectly intelligent, nice functioning members of society and when Ds went through a stage of getting up at 4.30 every day screen time saved my sanity.

I think you need to unclench a bit, and leave him to parent his way. You’ll soon be dealing with two kids and a bit of screen time will be necessary now and again! The much bigger issue is his attitude to your pregnancy, frankly.

I wouldn’t be bothered about screen time. I would be VERY bothered by this:

C) he has absolutely no idea how tiring growing another human is (tips for how to explain this to him would be helpful; I've tried showing him articles and he queries whether anyone has scientific evidence for my claim).

This man is disputing that being 9 months pregnant is exhausting?

I'm generally in favour of no screens, but if you have toddler who wakes at 5am and you've been working late the night before you do whatever it takes to get through the day!

Literally nothing wrong with screen time, especially in circumstances such as yours

He worked late last night, is working this weekend and you’re heavily pregnant and both up at 5am?! Screen time is your friend!

You can’t dictate how he parents, honestly it sounds like screen time is absolutely okay in that situation - it’s lovely you were doing sticker books etc but he doesn’t need to do exactly as directed.

Children do not need educational activities at 5am.
I think you are just making yourself stressed.

Screen time ALL the time is bad parenting, screen time when one of you is heavily pregnant and the other is trying to hold down a job around limited sleep is when it should be embraced.

Why couldn't you rest while you're DH was downstairs with your child? I'm confused

Why do you think you should be able to dictate how he spends his time, I assume you wouldn't like him to do the same to you? Kids don't need constant entertaining