Muddling If I knew what I know now at the start of things I would have done a better job I think.
I think we all would. Everyone has regrets, but it doesn't help to dwell. You did what you thought was best at that moment in time.
Runner I'm sorry DD didn't manage today after a successful day yesterday. Has she spoken about yesterday? If the school can tell you now they would offer a place based on yesterday if DD attends tomorrow and Thursday then I would push for them to offer a place regardless of what happens over the next 2 days on the basis that starting properly allows for a slower transition. You have nothing to lose by pushing this point. DD has shown with the right support she can engage. Have you looked at out of county schools?
Being full (which isn't legally defined) isn't enough of a reason to refuse admission. The LA have to prove they are so full it is incompatible with the efficient education of others or use use of resources. The bar is relatively high.
Try to get an appointment with IPSEA or call SOSSEN. IPSEA are trialling a call in helpline on Fridays, too. IPSEA information on their website is a starting point. And, SOSSEN have booklets and webinars on appealing you can purchase. You aren't eligible for legal aid by any chance? You need to appeal B&F as well as I as they will be written to support the placement the LA name in I.
Cantonet that's a cop out, don't let CAMHS discharge DS. They can help, don't let them tell you they can't.
Always DS1 was 5 when his MH difficulties started. At 6 he was diagnosed with PTSD with associated anxiety and depression. He then reached crisis point and started attending a CAMHS day unit. DS3's anxiety became noticeable at 6/7, although his anxiety is related to his ASD rather than a standalone problem. With appropriate autism support it isn't a huge barrier.
I think you have to push DC to an extent - as Bubbles posted, enough to push them out of their comfort zone, but not so it's detrimental. Sometimes DC can't rationalise what is in their best interest. Can you make leaving the house part of fulfilling functional needs. For example, saying I need to go to the bank/walk the dog/post a letter/take sibling to school and you can't stay here on your own. Or I need your help to wash the car/weed the garden/paint the fence/rake the leaves/carry the logs... Once DS is leaving the house every day to fulfil functional needs you can work on more. You can make an exposure ladder - in the garden/go to the end of the drive/the end of the street...