Well, if nothing else, the concepts of “traditional” and “conservative” are applicable to the horseshoe.
I might offer the advice to figure out what your bald clients do to keep their heads warm. After all, they’re the membership of the club you’re looking to join.
No, the beret is a bad idea. In all seriousness: my vote is with the flat cap/driving cap/golf cap/gatsby cap/newsboy cap/Great Depression cap/whatever the hell else you want to call it. This is definitely the best hat if you’re seeking to both be dressy/classy and also unobtrusive. A solid-color one would go well with a dark suit and coat. It hasn’t been appropriated by hipsters yet, so it doesn’t have the same “look-at-me” vibe as the fedoras, trilbys and other brimmed hats like that. Old guys at my Masonic Lodge wear flat caps (accompanying formal dress and overcoats) without any ostentatiousness whatsoever.
ETA: the key with this type of hat is that it has to fit properly. One that’s too big will look goofy and floppy.
My take is that, when it’s cold out, nothing that you wear outdoors to keep you warm counts against you until after you have come indoors and had a few minutes to prepare to present yourself.
So I’d go with the (easily squashable) knit hat, taken off (stash it in the briefcase or something) as soon as you are indoors.