A friend recently visited the Theodore Roosevelt Center in Dickinson State University's Lowman Walton Hall in Dickinson, ND
https://www.theodorerooseveltcenter.org/

Roosevelt's L.C. Smith Maker of Baker Gun, Syracuse 3 barrel drilling is on display
https://www.theodorerooseveltcenter.org/Research/Digital-Library/Record?libID=o277317





The gun is not mentioned in Houchins’ but it is very likely listed in Hunting Trips of a Ranchman published in 1885 as his “ranch gun”:
“I have two double-barreled shotguns; a No. 10 chokebore for ducks and geese made by Thomas of Chicago; and a No. 16 hammerless built for me by Kennedy of St. Paul, for grouse and plover. On regular hunting trips I always carry the Winchester rifle, but in riding round near home, where a man may see a deer and is sure to come across ducks and grouse, it is best to take the little ranch gun, a double-barrel No. 16, with a 40-70 rifle underneath the shotgun barrels.”

The L.C. Smith 3 barrel guns were manufactured 1880-1884

By the engraving and hammer sculpting it appears to be a Quality No. 3



Tough read but I think the barrels are 3 Iron "Oxford"