Hi Chris.
My research results and my theory is like follows:
The shown coats are regular coats of the old days. The coats were worn by the shooters, doctors, teachers, pharmacists, factory owners, and, and, and. It was a good quality wool coat for the wealthy society. Often shooters had their honor decoration badges and pins on the coat. Or the fabric ribbon across with the colors of the country or club, especially they convoyed the banner flag of the club in a parade.
And those coats existed all over the German states back then and were brought also to the new world. It was just a pretty coat, not a kind of uniform.
The Prussian shooting clubs back then had more a kind of uniform, which they wore for parades and club events. They still wear it nowadays, with a hat and their fabric-badges stitched on the sleeve.
If you have a look on the homepage of the Brooklyn or New York Schützen-Corps, you can see their uniforms. What they wear is typical Prussian shooter uniforms. They are more the "society clubs" not real shooting clubs. They think they keep German traditions, thats their mission. And they mention their origin: North Germany (Prussia)
Old clubs with the name "shooting society" or "Schützen Gilde" were more "real shooting clubs", and often they are still nowadays at ranges to shoot their matches.
I don't know how many real German Schützen clubs survived in US, after it was forbidden to be very German (during WWI). Nowadays there are still a lot of Bavarian clubs in US, which care for Bavarian traditions. But I don't know if there are also Bavarian shooting clubs.
There is no special Bavarian shooter dress. Nowadays they wear leather-hosen and wool coats in Bavarian style. But in the old days, it was much to expensive for a farmer or worker to own a leather pant. They wore regular black pants and a black coat. In the big towns like Munich they didnt had a kind of "Tracht" (costume dress). Munich shooters were wealthy people, company owners and so on. They wore a black or grey wool double-breasted coat with two rows of buttons, always with a vest and a hat.
The figure shows a bunch of wealthy high society shooters in 1863 (Munich), posing for a group picture.
Biggi.