If shooting in a vintage shoot in America, why look like a Brit? Not hard to reproduce c. 1900 American attire. Most competitors wore a shooting sweater in the winter. Destry had some "shawl" shooting sweaters made.
Tom Marshall, winner of the 1897 and 1899 GAHs
and a shooting coat otherwise; maybe lightweight linen in hot weather. George Work, 1894
April 27, 1895
Sporting Life
https://digital.la84.org/digital/col...id/47962/rec/1
Nicely dressed gentlemen c. 1915
I've seen very few illustrations or images of sleeveless shooting vests. Capt. A.W. Money and his Parker
1918 GAH
https://books.google.com/books?id=U_...=RA1-PA278&lpg
The Brits didn't wear kilts in trapshooting competitions; the 1901 Anglo-American match British team. And this was July
The 1908 Olympic British Gold Medal team
Not trying to be jingoistic, but
why in the world would any of us want to look like someone we are/were not??
A random dork in a boater

Boaters were very popular in the teens and 20s and make a nice hot weather accessory; esp. a classic straw "Skimmer" with a black and red band for an Italian
https://www.villagehatshop.com/hats/Italian-Boater