I hope this helps.
I'm sorry to hear what's going on with you, and it sounds like a LOT!
I'm kind of confused as to what point in the UC claim you're up to at present - if you've only just gone on to the benefit are you not still waiting for the Limited Capability for Work assessment?
Bottom line is, it's not the Work Coach who decides whether you have Limited Capability for Work, or Limited Capability for Work-Related Activity (Support Group), it's the decision makers at the DWP who decide that. What the Work Coach CAN do, is excuse you from looking for work while this assessment is pending - do you think this might be what has happened?
Alternatively if you have been migrated straight from ESA, a Decision Maker may have already placed you in the Work Related Activity Group?
You need to find out the answer to this, as if a decision HAS already been made, this is appealable, if it has not, then the question is what to do until a decision has been made - in this case, the level to which you are excused from work-related activity is at the Work Coach's discretion, and if you feel they are not acting appropriately I would start with a complaint.
You can ask your Work Coach to clarify this for you.
As I said, it sounds like you have a lot on your plate at the moment - if you want to pick our brains on anything else please do - and please also let us know how you get on with this.