I don’t live in the Bible belt. I live in NYC, where Catholicism is more popular, due to being immigration central.

So, I’m somewhat more optimistic.
I think you could innocently ask them, “Would you happen to have a Catholic Bible?” Or “Would you happen to have a Douay Rheims or NAB bible?”
If they say no, then just say, “Oh, I would be interested in purchasing one of those Bibles and your store is nearby my home. Do you think perhaps you could order one for me?”
Some bookstores, at least regular bookstores, are willing to order books they don’t have for others. You can definitely try this in places like Borders, Barnes and Nobles and the like. independent bookstores are likely to do it if they can charge you the shipping fee, or if they see a demand for it.
If they give you a judgmental look or get rude with you, just say, “Sorry” or “Thank you for your time,” and go on your way. If they don’t want your patronage, then it’s their loss.
But you know, I’m disappointed that even Borders by my church doesn’t have the Douay Rheims Bible. Has other Bible that are put in the “Apocrypha” section, but not DR.
Ebay and Amazon are just wonderful, IMO. Oh, and
Half.com, which is an extension of Ebay.
I hope God gives you the strength in an obviously difficult area to live in.