Just give the soldiers refurbished Cavalry muskets, except they'll need to be re-textured. Because the muskets were hand made, the artisan who made it usually added extremely intricate designs.
Also the frame of the musket itself was extremely different. A much narrower stock, smaller barrel. Much lighter than their European counterparts but extremely more expensive, plus they broke easier.
Last pic is a cavlary musket from 1844, but the design was the same. Carbines weren't used in the Army, only full length muskets.
The hard part is finding actual uniforms, there were no real codes other than "Have a gun and an outfit looking similar to the other people"