"I don't think he's bonded with our DD2 at all and he actually cried when she was a couple of days old and said he wanted her to be a boy".
What a horrible thing to do.
Was he negative prior to the birth of your first child or did this start after birth?.
Is he going to continue to ignore his DD2 because she is not a boy like he wanted?!. He sounds very much like someone who wants his own way, is so very emotionally immature and understands nothing about conception. What are his parents like OP; what did he learn from them about relationships when he was growing up?. You've recently given birth to another human being; what is he doing here about his supposed poor state of mental health other than blame you for it (which is wrong on all sorts of levels).
Re your comment:-
"I asked him to help me this morning by letting me catch up with sleep and him take DD2 for me whilst I napped but he threw a fit (and I mean a fit I've never seen him like it before) saying that he had planned to go out to meet his parents with DD1 and it's the only thing that's been getting him through the week because his MH is so low and I'm pushing him to the limit and making him feel like he isn't good enough"
There are red flags here re this man. You've recently given birth to another human being; what is he doing here about his supposed poor state of mental health other than blame you for it (which is wrong on all sorts of levels). What is he doing to improve things for his own self?. I am wondering if his negativity is a by product of him actually being an abusive individual, it seems too that he controls the moods of you and your household.
And no you are not stuck there long term either. I am wondering if he is feeling low (and I would argue more like completely pissed off) because he is no longer the Big Number 1 I Am King of the Castle in your life because your focus now is on your two children. His behaviour is also a probable cause of you feeling low too. Consider also going forward what you want to be teaching your daughters about relationships; would you want them to be with someone like this man?.