Not a 1st amendment issue
A decision by the U.S. patent office to deny trademark registration has been challenged by the owners of the Redskins football team and seems to be headed to the U.S. Supreme Court.
In 2014, the patent office canceled the Redskins' trademark registration, citing a law that prohibits trademark protection for names that "may disparage ... persons, living or dead, institutions, beliefs, or national symbols, or bring them into contempt or disrepute."
We believe that people have the right to use the term "Redskins," even though many people find it offensive. We also believe that a sports team has a right to call itself "Redskins." Denying a trademark does not deny anyone the right to use a particular term, it just means that the government is not giving official status to the term for brand identification.
A government agency making a subjective judgment on what is disparaging is not ideal but sometimes cannot be avoided. In this case, the patent office made the right decision.