Not read the full thread, just your posts.
Agree that childcare seems like a good idea but I think its necessary to have a frank talk with him about his attitude, which must be nipped in the bud now. Don't back down on this.
You need joint and fair expectations of parenting and its responsibilities, he doesn't get to duck out of them because he's now getting paid more.
You encouraged him for this job and were happy to support financially if things didn't work out, he needs reminding of this.
You're a team that supports each other, not a team designed to support one person and one person only. Your family and family isn't managed like a team at work. He needs reminding of this.
Without such a frank discussion you're making rod for your own back (and we'll see you back here in a few years needing advice on what it means for your marriage that your partner delegates all actual parenting to you).
If he's stressed about his new job, there are ways to deal with his worries and he's as entitled to request flexibility of some sort from his employer. (Let's face it, he'll have more luck with thos than if he were a woman!). Men are always happy to ask for more money, with such confidence there's no reason why they can't expect acknowledgement that they're a parent.
I don't mean this to sound man-bashing btw, I'm aware it reads that way! Everyone has legitimate fears and worries going into a promotion but it sounds like his response has been to take the easy way out - to put more responsibility on to you rather than actually deal with it. And deal with something that may well only be an issue in his mind. I'm sure you have faith in him that he's able to raise his concerns and deal with them in a practical way rather than being scared of talking to his employer.
Would you let your child duck out of facing something like this when they're older? I'd imagine that you'd help them to find a better way to confront their worry that leaves them stronger and this is the supportive attitude you can take to your husband.