Limited capability for work advice

So I currently work 14 hours a week, my boy has just turned 3 so now they want me to work 30 hours a week. This is not possible at all as I have no family support or anyone who can help with looking after him or picking him up or dropping him at nursery. On top of that he has suspected autism plus I have a physical disability and am struggling severely at the moment with my mental health and am under a mental health doctor. But UC will not back off, my work coach yesterday wanted me to leave my current job to do a full time unpaid training course. They keep telling lies as well, they said they cover the first month of nursery but it was a lie as I already work and now I am stuck with a bill I literally can’t pay. I was looking at applying for limited capability so they stop pressuirng me and causing me more stress but don’t want to leave my current part time job as it’s helping my mental health but I don’t know if I can apply while I’m working? I am happy doing the 14 hours I do but anymore that that will be to much for me to handle mentally and physically.