talk to the person who handles the billing in the doctor's office. Explain the situation (that you don't have the card because her father refuses to give it to you). See if you can do one of the following situations:
1) If you know the insurance company, have a call made to request the information. The billing person can easily make this call (I've had them do it before to find out if my insurance covered certain procedures) and it's likely that they can get the information they need faxed to them.
2) Have the bill sent to the father. If he's not giving you the insurance information, have the bill sent to him so he can take care of it himself. He'll either end up doing the work himself or make it so that you can just have everything billed to the insurance automatically.
3) Pay for it out of pocket, get a receipt, then send him a receipt of your own requesting payment. Keep the original receipt as a proof of payment (in fact, pay with a check or credit/debit card so you also have bank records). If he fails to pay you back or report it to his insurance you can take him to court.
Also, just a random thought, has your daughter made an insurance-covered visit to any other doctors at the same hospital? If so, the one you take her to will probably be able to obtain the insurance information from the other office. I had that done after my son was born. I didn't have my insurance card on me for his first visit, but since it was billed to the same people under my name the pediatrics office just pulled the information from my file from my OB file just fine.
Good luck =)