Oh OP, I was in almost exactly your position a couple of years ago.
Except I was expected to do a similar length round trip three days a week. And I was expected to start this from the moment I was safe to drive after my c section.
I also had a terrible-twos year old as well as a new born to deal with and, to be frank, I just didn't want to spend two hours in the car with them three days a week doing the school run. And in our case, we did have other help options, but DH didn't want to use them because it would have meant asking DSS's mum to step up a bit more (backstory alert!) and he didn't want to.
I posted about it on here at the time and got my arse handed to me because I was 'pushing DSS out' and I obviously hated and resented him and didn't want him to know his siblings or be part of our family, blah, blah, blah, usual step mother-hating crap on here.
So, I did it. For three days every week I did two hours in the car on the school run, recovering from a c section, with a tantrumming toddler and a newborn baby who SCREAMED all the way.
It was hell. I got PND. I used to drive all the way there with tears streaming down my face as my baby screamed in the back. Boobs leaking. Once I'd dropped DSS at school, I'd drive round the corner to the Tesco carpark and feed the baby in the car while the toddler gave out in the back and I sobbed.
It nearly broke me and it honestly nearly broke up my marriage. DH was a knob about the whole thing. Thought that 'sitting on my arse in the car for an hour each way' was 'easy' and couldn't understand what my problem was. He twisted every single thing I said to make out like I was saying I didn't want DSS to be here. Wouldn't even consider my point of view at all. I'm getting angry thinking about it all over again.
I limped on like that for a year and then went back to work and since then, DSS's school runs have been entirely the responsibility of DH and his ex to sort out between them. I must admit I do enjoy the fact that DH finds it difficult to juggle DSS's school runs - and that's without having had major abdominal surgery, a toddler and a breastfeeding newborn in tow!
Please put your foot down OP. For your own sake. Fuck what everyone else wants or thinks at this point. You have a brand newborn baby. That, and your own physical and mental health must be the priority right now. Revisit things in a few months or so, once you're a bit more settled.