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I do not know why your insurance is asking for the school insurance information, since school insurance coverage provided is considered secondary insurance meaning they would be liable to pay what your primary insurance did not pay. However, you are entitled to file a claim with the school district for anything your insurance did not cover and you can provide the information to your insurance for them to try to file a claim.
The only way you would have an actual legal claim against the school would be if their insurance policy states it is primary coverage, which would be very odd, or if the injury occurred due to negligence of a school employee in which case you would have grounds to sue the school for negligence and they would have to cover if the court finds the injury could have been prevented if it was not for that negligence (as you mentioned).
You are entitled to file an insurance claim with the school insurer as a secondary claim and the school has to provide you the insurance information to file that claim.
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