The sum total of what I know about the subject is a book written by Mike Fortune-Wood, which I find persuasive. It is called "Can't Go, Won't Go: An Alternative Approach to School Refusal." Fortune-Wood is a long-time home educating parent with a particular interest in school phobia / school refusal. He has provided voluntary support to other home educating families for more than a decade.
In his book, he says that in clinical and educational circles there is widespread acceptance of the idea that allowing distressed children to leave school and be home educated will cause them to become more withdrawn, isolated and disturbed. However universally accepted it may be, this idea appears to be a longstanding professional myth. He has been unable to identify any research whatsoever which supports the view that home education is harmful to children such as your daughter.
If I remember right, he does not have any clinical evidence that home education is helpful either. He does, however, present several dozen case studies in his book of children who have been home educated following school refusal. Not all of the children continued in home education in the long term, and not all of them were "cured" completely of their mental health problems by leaving school. But without exception, all of the parents were of the opinion that home education had been a very positive experience and was just what their children needed at that stage in their lives. The parents reported that far from becoming isolated, after leaving school their children showed an increase in confidence and began to develop an interest in being with other people in situations they had previously found unbearable.
If the view of the CAMHS counselor has any influence over you, I would hope that is because of his or her professional knowledge of the subject. I suggest you have a look at the book I mention, and then ask the counselor firstly to direct you to some research which indicates that home education is harmful to children such as your daughter, and secondly to tell you how many home educated children he or she has worked with directly and whether those children have got worse after leaving school. By "home educated" I don't mean children who are temporarily off school due to school refusal, illness or having been excluded. Such children may not be physically attending school but they are still living under the constant threat of being forced to go. I mean children who have been taken off the school roll and told they do not have to return until and unless they feel ready to do so. Most parents describe this as a key difference and say that their children only relax and begin to recover once they are free from the fear of school.
Don't be fobbed off with vague assurances such as "it stands to reason..." or "it has long been recognised in clinical circles..." Many beliefs in this world have been widely accepted which have turned out to be untrue: it seems obvious that the earth is flat, that the sun goes around the earth, that home educated children will be lonely and will develop poor social skills. If the counselor's opinion is just prejudice or speculation, then it is worthless to you. If you are going to trust somebody's uninformed intuition about the situation then you are better off trusting your own rather than the counselor's; after all, you're the one who knows your daughter best.