There's the danger of mercury poisoning and if you were to only feed canned tuna to your cat, they wouldn't get all of the nutrients they need. This is very unhappy news for Juniper the Cat, who only has cans of tuna on her Christmas wish list.
I did some hunting and found nearly 20 different businesses in Lubbock that allow you to bring your pet inside. Some are chains that have a company-wide policy allowing pets, but others vary from store-to-store. So, I double-checked and called any that were uncertain to make sure.