For anyone who doubts that authorities go after parents who challenge, they absolutely do. This happened to me, the thing is, holding professional people to account they take as if you're actually trying to lose them their job when all you're doing is trying to get the best for your child.
OP hold your nerve, keep calm & go & see your GP. There is something you can do I think you could get an assessment that will say that you have capacity. You can ask for a capacity assessment,? I don't know if you'll get one but worth a shot. Or just try to ignore these allegations, they are distracting from the main issue. It's abusive behaviour, gaslighting & making you out to be insane.
You could write a clear letter separate from your child's information, calling them out & inviting them to assess your competence. Make it clear that your GP believes that you have capacity & are not ill (get their support first). But also make it clear that this will be the final time you will be addressing the issue, that it's a distraction & that your mental health is not up for discussion. Even if you had mental health issues, so what anyway, it's none of their business & they are showing themselves to be uneducated if they believe this is even an issue. They should still be trying to help your child either way.
I'd also consider advocacy? LA's usually have a linked service.
I challenged camhs & they sent a very very long letter to my local authority strongly implying I was deliberately trying to hurt my children, like fictitious illness. They basically ignored the fact that DD was at the time starving herself (ended up as an emergency going into an eating disorder unit). Fortunately for me, we were in a different LA to the camhs department, I say fortunate because I'm 100% certain that they had a relationship with their own child protection department.
It was lucky for us that we lived in a county that rejected the referral, but the writers of the referral letter (camhs,) tried to shut me up by destroying me as a parent.
There were no real concerns about me, they were making up stupid reasons to say I was trying to hurt my child, literally things like "using the kitchen" which they said was inappropriate.
Absolute fuckers. It left me with a real fear & mistrust of these authorities, honestly. They shut up in the end but it was hard being labelled & everything I said was analysed & misunderstood. I honestly felt like the therapists at that camhs were mentally imbalanced themselves. One of them is now quite high up. Yeah makes you wonder how these people are chosen.
But going back to OP, hold your nerve, try to get some advocacy, directly address their "concerns" go to your GP (they will have seen this all before,,), also keep going with getting your child help & support.
If you want to DM me please feel free. I'll help if I can.