On the contrary, having had what I suppose could be called 'mental health issues related to gender', or 'gender dysphoria' as a child, if that is an acknowledged thing, I can tell you that the least useful thing would have been to 'pander to my delusions' .
It is cruel to tell a child that if they keep saying they are the opposite sex, if they keep insisting that other people accept them as the opposite sex, they will actually be able to change sex - holding out the possibility future of non-medically-necessary chemical and surgical interventions at worst, or living their lives in a questionable identity at best.
The term 'transgender child' is inappropriate - it is sticking a highly contested, adult label on a child who deserves to be treated as an individual little human being who is either a boy or a girl, and who will inevitably grow up to be either a man or a woman.
'Being a man' or 'being a woman' does not have to be limited to the gender stereotypes that the trans movement has reinforced - a gender-questioning child can grow up to be a gender-non-conforming man or woman. Labelling them 'transgender' at a young age is forcing gender stereotypes onto them - if you're not a girly girl, you must be a transboy etc.
And yes, it does imply that there is something wrong with their bodies, so although the trans movement has 'retired' the 'born in the wrong body' thing, probably because it was so obviously at odds with the body positivity movement, the implication is clear: a little boy who believes he is a girl, or a little girl who believes she is a boy, have an obvious disconnect with the inescapable reality their physical bodies.
What gender-questioning children need is loving, caring, understanding but fact-based support to help them accept that they have been born with just one of the two biological sexes, and it is impossible to change that - now how can we lovingly support you to grow up in the sex you were born with, free from stereotypes, loving yourself as you are, and the body you will always have?
So the reason given for rejecting this child - “Our decision was influenced by the fact that we had accepted a family with a transgender child for that camp" -
is based on the damaging idea that a child can be 'transgender' - no, a child can be gender-questioning - 'transgender' is an adult label that damages children's freedom to grow up beyond gender stereotypes.