"Thing is it isn't DHs fault I'm a pushover" actually it is - because you're being a pushover because you're sleep deprived because HE isn't stepping up to his responsibilities!
Are you bf? Even if you are he needs to at least:
Do more of everything else baby needs when he's home (baths, changing nappies, comforting/amusing baby, rocking baby to sleep).
Do more of everything else! He's got more energy so he can bloody well do more!
Be responsible for baby late at night/early morning so you can catch up on sleep then. Eg could you go to bed 8/9pm and if bf dh can wake you only when baby needs fed?
"I've just told my DH that I feel suicidal. His response was 'that's really not advisable'. I then told him I have had an hrs sleep tonight. He told me I was wrong and I've had an hr and a half. I had to go through and explain why he is wrong" wow! Selfish heartless dick! I'm raging on your behalf! 
Have you anyone can speak up for you? Where's your mum? If my (now ex) had been like this my mum would have gone through him!!
If not mum then health visitor? Dr? Friend?
If I were your friend and you told me this I'd have no problem giving him an absolute bollocking!
AS SOON as there is someone awake who can take baby hand baby over and get some sleep. And if nobody is awake by 7 I'd be kicking bastard h's arse awake!
Going forward h needs a kick up the arse! And practice putting baby down (warming their bed and perhaps putting clothing near them that smells of you can help. You CAN NOT go on like this you will literally go nuts!!!
I also (safely) co slept a lot made things much easier.