how about negotiating with the younger sibling and giving it a try ... say ... 3months or 6 months ... chances are that this child WILL find friends quickly and like the new school but if not ... you can return to the old school ... less convenient to have two kids in different schools but they won’t kick out the older child out of the private one once they are both accepted and with the money saved you can arrange after school care or transport if needed.
I WOULD at least insist on a trial ... I am not one to torture my kids and both of them are emotional and sensitive ... if they don’t like their school I WILL find them another option but I do make them try something that I think might be good for them . It is a compromise for all involved ... trying out a school for a few months or even one schoolyear will not be super dramatic for a young child ( and believe me ... my kids had the worst seperatiin axiety ) ... I worried myself sick over every school change and what happened every single time was ... tears the first day, an ok second day and then ... new friends, a wonderful item from the treasure box or an exiting playground and ... child was just as happy as before ...