I'm going to go slightly against the flow here and stand up for your DH, OP.
The year after my first DC was born, my DH suffered from depression - effectively he was the one with PND. He was terribly distressed, miserable, tired, anxious, sleep deprived and of course I felt incredibly sorry for him and did my best to be supportive and helpful and patient.
But as the partner of someone with depression I can say that it is awful, awful, awful to live with. The unending misery and complaints, the absence of a pleasant, positive conversation. The interpretation of every word and action and event in the most negative way possible. It is exhausting and I don't think that it is fair to advise that the person living this environment should be expected to soak it up endlessly without feeling unhappy or frustrated themselves. Ok, I understand that my DH was deeply unhappy and in mental anguish. But for a lot of that year, the parts of my life I spent with him were also ruined. A partner's feelings are not as unimportant as are being made out in this thread. I can entirely relate to his "don't do this to me" comment because after a few months of DH's misery, the mere thought of him starting
I don't say this to make you feel worse, OP, but to let you know that actually your DH might just be a very nice human being who is also struggling to find the right way to cope with a difficult situation. Okay he might say the wrong thing from time to time. But then he is human, no? Don't write him or your relationship off - try to be kinder to and more forgiving of yourself, but him too.
It does also sound to me that you have martyr tendancies, which are adding resentment to the swirling mass of negative emotions you already have courtesy of your PND.
However, I agree that he sounds rather useless, but in the spirit of kindness/forgiveness, I think it is possible that he is less efficient and tuned in to the practicalities than you are because:
(a) you do this much more than he does, and have had time to learn from mistakes, work out your own routine, ways of managing your two DC etc; and
(b) you take over from him whenever he gets it wrong, thereby relieving him of the responsibility and the learning experience.
Ok it would be nice if he just knew what is needed, but he doesn't. Now is the time to be absolutely straight with him and agree in advance the basics that you need him to do. Eg, every time you both go out together, it is his job to dress DC/put together supplies/put the pram in the car. Every time you come back in from a joint outing, he will bring the pram back in from the car. There must be loads of regular tasks you can delegate? I bet he would appreciate just knowing what is expected than constantly upsetting you. Think about it first so that you can get a much content into the one conversation rather than feeling as though you are constantly nagging.
Also, you need to put your foot down more - you really do know best. If you think that a meal out would work better with a Bumbo, then insist. Until he has as much experience as you, your views about the practicalities are better informed and more valuable. But, if your DH ultimately does something wrong, always always always let him cope with the aftermath rather than sorting it out for him and silently seething.
Also I am 
at your breastfeeding exhaustion at this stage; it sounds extreme. Please get your thyroid and iron levels tested, what you describe isn't run of the mill. And also make sure that there is enough carbs/sugar in your diet to keep you going. I had an extra meal every day that I breastfed - normally tea and cake mid-afternoon. Without the dairy milk sugar in my system I really flagged.
It strikes me OP that you and your DH love each other but are both floundering in a difficult situation. Some bossy groundrules will give you both clarity.