"To all of you saying zero chance that's not accurate. You don't have to be approved by SS or anyone to become a fosterer with a 'private arrangement'."
It is now only family, or a very close family friend that would be considered for "private fostering", if resident in the same country, so I would think that the OP wouldn't be considered.
Any plans would have to be long term, given the age if this child and the circumstances, it wouldn't come under the existing "Private Fostering" legislation, which is due to be extended and made more protective of the child.
Why should she be considered over an established, experienced Foster Carer, once the family arrangement has ended (if it ever does)?
OP, you mention "Institutions", children don't go into institutions anymore, you seem to not have any understanding how the whole system works.
There are probably plans being made. Your intentions are good but not thought out fully.
This would be a full Fostering SS arranged placement, which would mean SS involvement/assessment in your family life, including extended family members.
I doubt it would be approved anyway, because you have no connection with the family, other than your child attending the same school.