In essence, you are a single parent of TWO. He is worse than useless, he is an actual burden on you, taking time and effort (and probably money) that you should be able to put into your job to enable you to keep a roof over your head and food into the two mouths that matter, yours and your child's. Being a single parent of ONE would be so much easier. And probably cost you less overall per month, too. And would certainly be better for your own mental health, too.
This may sound like a silly question, but do you think he is really neglecting your child or is he just calling you and pretending he is in order to get you to take over HIS parenting responsibilities? Because if he is truly neglecting his child he is 100% doing it on purpose. And anyone who would do this is not a person I would want anywhere near ANY child, let alone my own.
What you need to ask yourself is what on earth does he actually contribute to your life? He isn't contributing his fair share of childcare costs, you've said you'd actually get more in benefits than he pays into the household, and he isn't pulling his weight in caring for his child. Plus you're having to run around putting out HIS fires and tiptoeing around his 'issues'. It's a wonder you haven't had a nervous collapse!
If you booted his sorry ass out, would you be entitled to child maintenance? And benefits? You may be financial better off without him than you realize.
Whatever you do, do NOT go on a nightshift unless you are 1000% positive you could go back on days at the drop of a hat. Because if you do and things go belly up, you'll play hell trying to find nighttime child care. And you already know that HE won't be doing it.
And if he really is having MH issues, then it would probably be better all round if he moved out and got the help he needs. Can he go to his parents?
You need to get your ducks in a row, figure out finances, childcare, and housing (ie whose name is the lease in, would it be easier for you and baby to leave, etc). Then tell him he needs to move out 'for his own mental health'. Or leave yourself.