Historical accuracy for an undertaker's garb would depend a lot on where you lived in the "developing" West, and at what time.
Up and down this valley, until quite recently, there were still at least three funeral homes operating in conjunction with a furniture store. That's because in the old days, the guys making and selling furniture also made coffins . . .
Up 'til relatively recently, close friends and kin saw to the preparation and "laying out " of the dead, so the fancy trappings one associates with Victorian "undertakers" such as formal clothing, top-hats with crepe bows, etc., etc. would probably not have been the norm in most small places, and not even an option early-on. Which is not to say there wouldn't have been a demand. For example, in 1864, when the Vigilantes hung the notorious Jack Slade, his fiery widow had a lead-lined coffin built, and filled it with whiskey so his body could be transported (by wagon) from Virginia City MT to Salt Lake City for burial. (his gravesite has been lost, BTW). It would probably have been another 15-20 years before the services of a professioal mortician would have been available in that area, though.
You can still see some very fancy hearses displayed in the big boom towns like Virginia City Nevada, and Tombstone AZ though, which indicates that funerals with all of the customary formality and pomp including being pulled by six black horses with plumes, and all were available in those places in the 1880s. Photos of the Clantons and McClearys Wyatt and his thugs murdered show them in very fancy coffins . . . no simple pine boxes for them !
Parting thought: the tradition of an abundant floral spray over the coffin had a "functional" aspect in the olden days . . . lots of flowers would somewhat muffle the ominous, final sound of clods hitting the coffin . . .
On that happy note, y'all have a dandy-fine day !!