I recently experienced a similar issue, but had a longer time frame to resolve, and AM email support fixed the name transposition issue.

In your case, if unable to resolve over phone, and with two U.S. travelers with no visa issues or other travel restrictions, I would have shown up at airport prepped to travel, and worked with agents to resolve, as name transposition on international tickets is a fairly common issue.

At this point your flights should have departed (today is the 19th, you were scheduled to fly on 18th based on your message) -- how did this end up working out for you?