Do they offer methods other than SMS?
It's hardly surprising that customer service agents will not tell a voice on the phone how to work around a security protocol. This doesn't mean that there does not need to be an exception handling process, but it wouldn't be performed by front line customer service. I would contact them again, during normal business hours (on a day other than Monday), and be prepared to work your way up the levels to someone who can fix it. I wouldn't ask them to explain (they should not), stay focused on fixing.
They don't unfortunately or I would have just verified by email.
I suppose I'll call back next week. Just to clarify, I wanted to contact the unnamed third party to find out the source of the error since the customer service rep was unable to provide me with any alternatives to resolve it over the phone. I'll try again next week and report back.