I always disinfect anything that I buy from a store in which live birds are sold or displayed, just as I would if I were to buy something at a bird fair. Animals sold in pet shops are not always in the greatest health, as they by and large come from large factory breeding operations, and plenty of people have bought animals from these stores and had them die or fall supremely ill shortly afterward.
Honestly, any time I go to the pet store to buy something, I spray myself down with F10 before I even go in my house. It's (probably) overkill, but I have many animals of many species and I don't like taking risks.
As others have said, baking at a low temperature or a bleach dilution will work fine to disinfect a branch.
ETA: what's the bird-safe disinfectant you have now?