You can appeal the court's decision, but not yet. Appeals are generally based on final orders. The order denying your request is not in the nature of a final order. Once the pleadings are fully litigated and a final order is entered, you could appeal then.
So, you would need to wait for the matter to be concluded. If you disagreed with the final order, you could then appeal and one of the issues you would appeal the court's decision in denying your request.
Your only options right now are to accept the court's ruling for the time being and move forward or you could file a motion for re-hearing. Such a motion would be asking for a second chance to litigate the request for records. While you can try, such motions are not granted often, after all, you would be asking for the same thing from the same judge that just denied you.
Additionally, should there be some change in circumstances in the future, you could bring the issue up again. For example, should the mother's issues continue or if they became more relevant for some reason, this could constitute a change sufficient to have the issue reconsidered.
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