[Adam]

With a 5th foul, we typically pregame that the calling official will inform the coach, as he's already there at the table. If he senses pending issues, he can absolutely hand it off to a partner and switch.
It sounds like, in Billy's game, I would have informed him as the calling official as the coach had presumably been a peach all game. I also would have likely ignored the "you missed a foul" comment, assuming it's the first such comment all game.
However, if I warned him (always possible depending on many unknown variables), the T has to follow if he ignores the warning. Otherwise, you can be sure any coaches watching (and paying attention) will start working me at tip-off when I have their game.