Oh boy, OP. I think that this time in one's life is sooo hard. Looking back over the years at periods of bereavement, tough diagnoses, infidelity, redundancy -I was never more desperate than when DC were babies / toddlers and didn't sleep well, like I lived my life with my skin peeled off; everything hurt and there was zero rest or peace. I absolutely get your exhaustion. I used to fantasise about the national roll-out of a service to relieve sleep-deprived mothers, and felt so invisible, limping along with this secret impairment, trying to keep up.
I too work similar hours, in a pastry-rich environment, and am tied to my workplace with no chance of WFH (you're not a teacher, are you?). It would take immense willpower to resist the sugary offerings in the addled state which years of sleep deprivation puts one in.
And the male partner who ought to do more but, even after all the discussions and arguments, can't be persuaded? I used to find it so disheartening when people said "Well, he just has to do his share," / "Just go out and leave him with the baby," / "He'll figure it out," / He'll have to figure it out,". Because, at the time, if there was anything I could have said or done which would have indicated that this was a realistic possibility, I would have done it already. Comments suggesting I should "just" made things feel ten times worse; as if I was stupid for not having done x or thought of y already as well as exhausted beyond measure.
A previous poster suggested a walking pad -that's a great idea. People say that weightloss is mainly diet and that you have to work out a lot to see a difference, but I'm sure people's metabolisms are different. I can restrict all I want, but unless I also throw in some exercise in order to get my metabolic engines running, absolutely nothing happens. With an opportunity to walk at pace and incline in your own home when your DC is asleep or otherwise occupied, even for just short periods here and there, you give yourself the gift of moderate exercise without leaving the house. My walking pad is the safety valve which keeps me on a healthy trajectory now. My DC are older and sleep isn't an issue because of them anymore, but I work late in the evenings and at the weekends so finding time for exercise can still be hard term-time (I work out religiously in the holidays now).
You'll emerge from this time at some point down the line and change will be easier to bring about, but until then, please be kind to yourself. You're in the crucible of parenthood, going through a tremendous rite of passage which is likely to change how you perceive the world, and your place in it. Make it count: this is the time for the inner work, the tempering of your empathy, compassion and courage (starting with yourself), and, once through the furnace and out the other end (when your DC fiiiiiinally sleeps better), you'll have the energy for the outer work you want to do. One foot in front of the other, OP. You've got this.