I don't think we should assume that the credit card Rotana used discussed by authorities is issued by an American bank. It is entirely possible/ plausible that the card was from a financial institution outside the U.S., from Saudi Arabia or the EU.
If the card was a high limit (which seems quite likely), it's possible it was opened by the father on behalf of Rotana, who would not have qualified for a high limit card on her own. Since the father was mostly outside the U.S., he may not have been aware of the rapidly escalating charges and hitting the limit. (Perhaps the father has an accountant who takes care of his financial matters for his family in the U.S.?)
Simply search for "Saudi Arabian credit cards" or Saudi Arabian credit cards with visa logo", Saudi Arabian credit cards to use in the U.S., or similar.
Of course, it would be very interesting to also know what the details are of the mother's finances since 2015-- how she had access to cash and credit, bank accounts, whether or not she paid the ordinary bills for the 3 kids and herself, etc.
Would US authorities have access to financial information/transactions on a saudi credit card?