The answer depends on the reimbursement rights contained in your health insurance contract. Most health insurance contracts contain reimbursement language (including from your own (UM) coverage), but some do not.
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Based on the facts you posted you could sue the driver (on behalf of your child). You should consult with a local personal injury attorney.
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While in California the CA Insurance Code does require an insured insurance company to provide an insured with written notice of the statute of limitations in uninsured motorist claims, there is no such requirement in underinsured motorist claims. You should consult with an attorney specializing in automobile accidents and personal injuries immediately to ensure that you have properly preserved your claim against the at fault party and for UIM benefits against your own insurance company.
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I would recommend that you contact other personal injury attorneys to see if any are interested in handling your matter. If not, you can represent yourself in small claims court.
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