Oh how I feel for you. I'm a SP but they're grown up thank god and they never lived with us. I don't think you are in any doubt that you ss is a very disturbed child and that's not of your doing - it sounds like he has never been wanted by his mother (and you have to wonder what his dad doing to compensate for this rejection.............) was there something about the child being looked after by his granny and then mom decides she doesn't want him............and you are having to pick up the pieces for all this. Obviously the child is hurting like mad and can't process the feelings of rejection and abandonment, and is insanely jealous of the new baby and in his eyes "mothers" are no good - they don't want you - and you're a "mother" so you're no good either................and you love the baby and not him. Sorry I'm not criticising I'm para phrasing what your ss is thinking.
Like many others I think the your DP is not being fair -at all - him and his mom have screwed up their child and now you are left to sort it out. Someone has mentioned the book "Parenting the Hurt Child" and it is an excellent book and used a lot by adopters, but if I read you correctly you don't want to "parent the hurt child" and why would you. IF your partner's attitude was different, if he was prepared to exercise care and control over his son and support you in your endeavours to cope with the child, maybe you would feel differently, but he's just sticking his head in the sand and making stupid excuses for his son's behaviour. I hate to say this but things are only going to get worse because if a child is messed up emotionally in his formative years (especially the first 3 years) that will cause problems to a greater or lesser extent, often through the lifespan.
CAMHS are unlikely to be much help. I'm a retired social worker and I worked extensively with foster carers and adopters who were coping with the sorts of behaviours you describe, perpetrated on them by their birth parents. Many fell under the pressure. I saw marriages break up, birth kids leaving home, mental health problems in one or both of the parents.
I agree with others that there is no way DP will want to end the r/ship if you cannot cope with his son. Sounds like he wants to be away from him as much as possible with this job. Someone asked about your housing - are you renting, or mortgaged. I think if you could afford it the best thing would be for you to leave DP and his son because (and please don't take this the wrong way) it isn't good for you or the 4 year old the way things are. He can't cope with having to compete with a baby and I hate to say this but there is a real danger that he could hurt the baby in an unguarded moment. DP and his ex need to take urgently about the child and make a decision about his care.
SO - if the only place you can go is your mother, then so be it - OR you stay and things carry on the way they are...............sorry it's tough I know. Look I might have this wrong but you don't sound like you are a very assertive person. I think with a DP like yours, you need to be assertive - seems he gets all his own way. You need to start standing up to him and laying down some ground rules.
Look after yourself too - and don't feel bad - I know that one - I had a terrible time with my SD and I used to feel like a monster for feeling so horrible towards a little girl, and a pretty one with long hair too!! I hid it but it almost cost us our relationship.