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In the absence of that employment contract mentioned above which we suspect you do not have, since almost no one does, your only alternative to wearing the costume is letting them fire you for refusing to wear the costume, and filing for unemployment benefits. Approval of those benefits would be a very big maybe/maybe not issue, particularly since you didn't refuse it all along, you agreed to wear the costume before, thus accepting the terms of employment and this being a requirement, and then now have decided not to accept it anymore because it's tacky and you don't like it. You live in an " at will" state. Your employer can legally fire you for just about anything and usually, in almost all cases, filing for unemployment benefits is your only redress.
Or you have the legal option of quitting your job for that same reason. Yours is also an "at will" state where you can quit a job at will. But your chances of getting unemployment if you quit are even slimmer.
Filing a suit against the employer? No labor laws have been broken unless you can prove a pervasive pattern of discrimination against people of your sex by management, which you have complained about to them and they have done nothing to correct. An EEOC complaint would take a really long time to investigate and process. If they find out you filed it, you'd very likely be terminated. Frankly, your best alternative to wearing the tacky costume is to find another job, right away, before you quit or file grievances, etc., Any attorney worth having would tell you that you don't have any legal case against your employer unless you could come to them with a "right to sue letter" issued by the EEOC, about a year from now (if you filed your grievance right away.)