They shouldn't be doing it, and it has nothing to do with whether there are native students or faculty. It's an outdated stereotype fueled by old movies and TV shows. It's offensive to everyone, or should be. It hardly ever is based on true cowboy garb or ethnic make-up (many were black, for example), and it usually is accompanied by Native garb that has nothing to do with tribes that were in the same states as those cowboys. Moreover, the stereotypes included natives saying "Ugh" and speaking pigeon English, and always losing to the cowboys after finding out their arrows didn't measure up against rifles.
I don't think this has anything to with conservative/moderate/liberal. I think it has to do with people who've actually been educated about the many native tribes and traditions, which vary far more widely than the stereotypical images.
Why not celebrate Black History month by having everyone dress as slaves or as black-face wearing minstrel show members? Why not celebrate Chinese New Year dressed like Hop Sing from "Bonanza" or saying "No tickey, no shirty"?
I think you should have your students write letters to your state's Commission Against Discrimination or the local chapter of the Anti-Defamation League to get some additional input. You should do a whole unit on the laws that have been put through to counteract systemic discrimination as well as the many government departments that have gotten in trouble by perpetuating these stereotypes.
There is nothing "just fun" about this. If they want to schedule a multi-cultural awareness day and have different classes or clubs really research different countries (geography, language, culture, food, dress), great. Our town's upper elementary school (grades 4-6) does this every year and it's very enlightening and done with the utmost respect and scholarship.
What you're describing is a poorly thought out version of a 1950s TV show, and it's going to reflect very poorly on the school. It's going to get a lot of negative press because someone's going to find out about it. The school administration and/or the sponsoring club is going to be in for a real fun time of making excuses and feeling embarrassed.