I don’t know where you get the idea that a hat is a begged isha. Until JFK decided to appear at his inauguration without a hat, that was what everyone wore.
I said Fedora. If you want to wear a pilleus cornutus or a Yarmulka to davening then Kol HaKavod.
I certainly agree with you that people do not dress with Kavod for davening, such as those who wear tee shirts (with a hat and jacket), and they should dress like they are “Omed Lifnei HaMelech”. I support your outrage.