I'll be the first to admit, just because someone is a doctor (or surgeon) doesn't mean they're smart in other areas. They can be a genius about medicine and an IDIOT about everything else.
That being said, what kind of dumbass open discusses their fake relationship in the workplace? When the people they're trying to fool ALSO work in that same place? Even if our villainess hadn't heard it, literally ANYONE else could have heard and spread the rumors.