You're coming across as very defensive.
Yes, boarding school is a choice. But it's often now flexi-boarding or weekly boarding.
Which does your DD do?
You say that there are advantages over a day school for you.
Are you willing to say what they are?
When I worked in boarding schools, most students were there either because of parental location (overseas), special needs not catered for in the state system, or state schools not being suitable for some reason.
Many more students now attend indie schools as day pupils. Boarding is on the decline and it's usually for parents whose lives are at a distance or they work crazy hours and can't do hands-on parenting. OR the child has a super skill (maybe sporting or music) that is nurtured more in an indie school. You know, Millfield for sports, Bedales for drama and arts, etc etc.
I know that if a parent had come to any school I'd worked in with a child's eating /weight issue, the staff would have bent over backwards to help. It's not about obesity being acceptable in the UK.
I'd ask you to think very carefully over whether your D is as happy as you think she is. Your posts show a lack of engagement with how her emotions may not be how you perceive them.
If it's not PCOS then it's more likely emotional eating.