OP, I do not have much to offer in the way of solutions. However, I have been in a very similar situation regarding my sons. I will simply relate the situation as an example of what happened in our family's case.
We have two sons who are close in age ( two years between them). My eldest has very obvious additional needs. My youngest masked, and any lack of progress at school or behavioural difficulties were assumed by school, by mental health professionals and by us (parents) as being the result of quite a challenging family situation.
My eldest son began having very significant mental health problems during his early to mid-teens. By the time he was 14, he was unable to attend school. Indeed, he was at significant risk at school because he would run away (into busy roads) and self harm significantly.
When he was unable to attend school, my husband had the sole responsibility of keeping him occupied and safe. This was not feasible, so I took long term sick leave from work to try to manage the situation. My eldest ended up as an inpatient in a CAMHS unit.
Then, my youngest began to 'refuse' school. There were interventions from the Educational Welfare Team, the Head of Year and eventually for social services to try to help him attend. He was diagnosed with severe GAD (Generalised Anxiety Disorder) and a home tutor was allocated from the LA to help him keep up with English and Maths. He hid from the tutor, locked himself in the bathroom when the tutor came and simply refused to engage. This was because he was overwhelmingly anxious, not because he was being defiant.
When my eldest came back from the CAMHS unit, we had two very unsettled teenage boys at home. There was very little provision for them, and we tried our hardest to encourage them to do something outside the house. It was Hell! The boys fought, my husband (who is also autistic) responded with violent meltdowns and Children's Social Care became involved at the highest level.
The rest of the story involves child protection plans and a necessary separation, largely due to my husband's behaviour towards the boys. This comprised lashing out at them, tirades of verbal abuse and breaking things in the house. I tried to manage the situation (sending my husband out; trying to separate the boys who were at each other's throats and so on). However, I did not do enough, soon enough.
The crisis happened ten years ago. Now my eldest is in supported accommodation and I hope he will start a little part time work soon. My husband was diagnosed with schizophrenia (co-occurring with his ASD) and is now in a care home. He is very unwell. My youngest is still chronically anxious and is being helped by a supported employment team to try to engage in some form of activities.
I think the main point in this oversharing is to say that the OP's situation is serious. It involves a family in crisis, long term inability to access education, a child with caring responsibilities (my youngest used to adapt his play for his brother, and my youngest is also neurodiverse), and, although played down, physical abuse of a vulnerable child.
My advice, if I am in a position to give any, is to contact CSC to see if there can be a multi-disciplinary team meeting in which all agencies pull together to help the children and to help the parents to cope.