Really sorry to hear you’ve been caught up in the Experian fiasco. Generally, what you describe is the result of one of two things.
The first is Republic’s numbers are Voice over Internet Protocol (VoIP). The FCC classifies VoIP numbers as landlines (wireline is the more correct telecom term). Although Republic’s numbers are perfectly capable of receiving text messages, some organizations refuse to recognize this. There is nothing Republic can do to change this regulatory classification.
The second thing that comes to mind is Republic is a prepaid rather than a postpaid service. Because prepaid numbers typically aren’t listed anywhere, it isn’t as easy to verify them as postpaid numbers.
Does the credit card issuer in question have alternative means other than receiving a text message to confirm your identity? I’m thinking they must as though it’s unusual these days, some still don’t own a cell phone.
Lastly, I realize this may not be very comforting right now, however, the most one can be held accountable on an unauthorized charge is $50. This is written into federal statute, specifically the Fair Credit Billing Act. More on that here: https://www.consumer.ftc.gov/articles/0213-lost-or-stolen-credit-atm-and-debit-cards. Speaking as a former banker, I’ve never seen a bank actually impose even the $50. It will take time and unfairly some effort on your part but this will get sorted.