I am with Wolf on this one. I have never seen any Rebbi or Rav kick someone out of davening, whether it be in a Yeshiva or a shul, if they weren’t wearing a hat and/or jacket.
Also, in my personal experience, whenever I daven Mincha in midtown Manhattan, were you have professionals of all makes and models, wearing suits of upwards of thousands of dollars, who meet distinguished individuals (secular and religious) on a daily basis, they never wear a hat. They were suits because that is their daily dress, but hats are officially out of style (even for showing respect).
I know that this is not a Rav and I am not going against R’ Chaim in the slightest, in a book I read entitled “A Gentelman Gets Dressed Up” written by Bryan Curtis (which I have found changed my everyday dress habits for the better) it states that in nowadays wearing a hat is actually taboo indoors (except if one is attending a jewish ceremony were one is expected to wear a Yarmulke).
I am just mentioning that society has changed from wearing a hat as being respectful to it being considered out of date and would not be used to greet anyone, especially Royalty and High government officials (and, as one stated, judge).