What should you do if you "call out" someone for being disingenuous, but then you get accused by that person of bullying?

There have been several times when I have pointed out that someone has stretched the truth, or attempted to create a picture of themselves that was untrue, most likely to make themselves look more interesting, or more glamorous, or more wordly, only to get accused of being a big meany by that person.

In these cases, I have only used the person’s actual words and testimony to point out discrepancies, and outright lies, but sometimes these people get so angry at being called out on their follies that all they can do is “shoot the messenger.”

Should the messenger just coddle the tall tale tellers?

What do you do when you smell a proverbial rat?