Anyone have any ideas?
Some more thoughts on this.
I do not believe that you need to have the same number of races from every dimension. Indeed, I think it would be just fine to abandon the Gringo/Indian distinction and have a single race "Human" that can come from any of your four dimensions and
is the only race that can be native to the Wild West.
If you want more "races" for the West, I suppose that you can use professional or social backgrounds where they don't conflict with classes - Cowboy, Miner, Englishman, Yankee, Mohawk, Cherokee, whatever. What mechanical differences you could introduce to distinguish these, on the other hand, is a very difficult question.