This is ridiculous--there is nothing remotely professional about this shirt. No reasonable person is going to say for a business that delivers this kind of thing is okay. First the business is not my Three *****es Cakes so your point about names is frankly irrelevant.
Second whatever you think of the shirt it includes a word that is considered profane. Oh right, as a noun it refers to female animals---if this is your justification--thank you for proving the point. This is akin to delivering a cake in motor oil. No wait, I would rather have a person deliver a cake in motor oil rather than seeing someone that thinks its good fun to refer to a woman as a *****.
There are standards of professional attire. And by your position showing up in a banana hammock would be just as acceptable- after all all standards of professional attire ultimately depended upon what some individuals have determined to be appropriate articles of clothing. Customers are well within their right to demand a refund on a delivery charge. If I am paying for you to deliver I am paying for a professional service as such I expect your delivery to reflect a standard of professionalism. And if someone who represents your business fails to provide that level of service you bet I am going to ask for you to refund what I paid you to provide that service. Delivery is more than simply showing up as you want as you nicely point out in your final line.
The OP has stated repeatedly it was inappropriate and was mortified when she found out her husband wore it....