DY: Who is “we”? Whoever is the “we” who views this as pikuach nefesh should supply the funding. No different than your medical case if the doctor refuses to operate without getting paid. Whomever views getting that specific doctor as pikuach nefesh should supply the funds, not the doctor who says go somewhere else. Personally, I don’t view Yeshiva as pikuach nefesh as IMHO home schooling is a real and viable option.
Without a real community school that has a mandate to take everyone, each school owner/administrator can push the child off and claim it is not their problem.