Well done OP.
There isn't middle ground here. It's separating the one child's needs out from the girls' needs and dealing with them as two separate things, as they are.
The one male child does not wish to change with the boys. Absolutely fine. The child's identity and preference can absolutely be respected, facilities can be provided, this can be done in ways that ensure privacy, dignity, respect for this child's feelings and choices.
The male child cannot insist on being permitted to change with the girls, because this cannot be done without infringing on the girls' privacy, dignity and respect for their feelings. There is no other way to put it but that this would be making use of the girls - regardless of impact on them, because this does not look at their individual and collective rights and needs - to confirm and support the male child's identity.
Support the male child - yes absolutely, and this can and should be done. No child should ever be asked to undress in a situation where they do not feel safe, or that others are present in ways that make them feel loss of privacy and dignity. This just applies equally to female children.
Moving female children (excluding) out of female provision in order to enable someone male to use it, is not an acceptable solution, it's sexist, it ends the female facilities doing what they are intended to do which is meet all female needs, and it punishes and shames girls for having barriers like faith, culture, disability, trauma, boundaries, a sense of privacy. It also pressures girls to put up and shut up to avoid that shaming and exclusion, and may subject those girls to division and questioning of who can stay and who can't, which also involves the male child (I don't know how the child prefers to identify themselves, so this is factual rather than disrespectful) in a way that pressures them. It's divisive, it's unhelpful, it's causing bad feeling where there does not need to be any, and no child benefits from it.
The messages here: there is nothing wrong with identifying as trans, and children who do are respected and supported. Safeguarding is taken very seriously, as is child wellbeing, so no child will ever be asked to undress in a situation where they do not feel safe, or that the presence of others affects their privacy and dignity. This does include girls. It is not possible to change sex, and changing rooms must remain sex based. However alternative private facilities will always be offered in addition where a child would like this.