Q: At what point does a professional person in a work capacity (especially when dealing with the public) no longer take the rudeness that these people dish out?
A: The point of measurement you seek is only able to be determined by you.
There is a difference in weathering a few challenges and being someone’s verbal punching bag.
Even athletic referees will penalize someone for direct assault insults to their character but they will take a lot of abuse about decisions in which another disagrees.