I think everyone answered your question pretty clearly. There's nothing wrong with a community rallying around a native son, or daughter, or team. However, when you're in a populated area with a large portion of that community being fans, or supporters of the other team, you have to decide how you need to handle it if you choose to ignore, or exclusively support one over the other. Pulls at people's strings. It's not rocket science.
The question I'd ask you, is while it's okay for an Auburn guy to order a pizza from that Pizza Hut, will it be possible for one of the bammer supporters in there to NOT say something about the Auburn customer's shirt, hat or whatever? I'd venture to say that 5 out of 10 times, they couldn't keep their mouth shut on the subject. I believe also, that 3 out of those 5 would say something disparaging, or something belittleing their support of Auburn over Alabama. Just a guess...and my opinion, but certainly based on experience.
I just really don't look at it like that I suppose. I never say anything to somebody wearing Auburn, LSU, or whatever gear. I was wearing an Alabama polo at the Shrimp Festival and an LSU fan said "Roll Tide" to me, so I gave him a "Geaux Tigers" back. I thought that was pretty spectacular for an LSU fan. Maybe he was drunk or something, who knows. The ACE Hardware in Foley flies an AU and LSU flag out by the street, but I still shop there. I often find that if I don't act like an idiot about football to somebody, they won't act like an idiot towards me. Maybe your problem is birds of a feather and whatnot. I know not all Alabama, or Auburn fans for that matter, can keep their mouths shut though.