there is no pronunciation difference. Peer is the only correct way to pronounce that capital. Just like if you pronounce Versailles, KY, vare-sigh, you are screwing it up. it's pronounced Ver-sales. it's a name, and the only way to pronounce it correctly is the way the person or place whose name it is, pronounces it. so you should change it to deer, or queer, or beer, or something else that actually rhymes. your ignorance doesn't make it correct.
same argument for Cheyenne.