While I definitely think there are larger offenses at some point it comes to a matter of respect for the organization that is employing you
While I generally hate comparing sports to the real world - because it isn't - this is one area where it almost makes a little bit of sense.
I have a buddy who works for a distributor of budweiser and other AB products ..... he is very loyal to the brands he sells and would never show support for another brand .... with his company if you were at the store and were willing a Miller shirt, even if it was just a nascar shirt of a sponsored driver or something of the like, he could get in trouble with his company
I work for a national insurance company - in our industry we definitely brand a lot of items ..... we have branded polos, laptop bags, pens, leather notepad holders, tablet cases, etc ......... it would not go over well if I were wearing a competitors brand while out in public ... I do have some old t-shirts and luggage tags that I still use from my former company, but would never take my old portfolio out while visiting customers or wearing the company face .... not a chance
If I did, not sure if I would get in huge trouble - but would definitely feel heat and have to spend time dealing with an issue that was pretty obvious to see coming down the pipe and very avoidable ....... would be a question of my intelligence and decision making more than anything
IMO, while maybe not a big deal CK would wear a dolphins hat the obvious ramifications and questions that would come from the action along with his response to it make me question how much he actually gets his role as the face of a franchise