I am assuming that you are the "secondary mom," as only that person would have genuine interest in having the child's claims paid.

Submit the claim information directly to the dad's insurance carrier. Call dad's company and talk to the benefits administrator. Tell that person that you need to submit a claim for an employee's dependent directly. If asked why, be succinct but clear that the employee is not being a responsible parent in this matter. Ask them to tell you how to go about doing it.

The benefits administrator should be willing to give you the form or contact information for you to do this. If she is not, go to the next higher level person with the same request. Someone there should step up to the plate to help you with this.