I think I agree. Fighting words of all kinds are already banned throughout the state, and racial and ethnic slurs would be included. I see no reason to single out any particular word for the special status of non-utterance.
(1) Words that "by [their] very utterance tend to incite an immediate breach of the peace" and are "directed to the person of the hearer" -- which is to say personal insults said to a particular person, and not just insulting words about third parties said in a general speech, on a billboard, in a book, and so on -- are indeed unprotected by the First Amendment, under the so-called "fighting words" exception. ...
(4) The ordinance is also superfluous. Texas Penal Code 42.01(a)(1) already makes it an offense if a person "intentionally or knowingly" "uses abusive, indecent, profane, or vulgar language in a public place, and the language by its very utterance tends to incite an immediate breach of the peace"; Texas courts have read this as limited to "fighting words," but of course covering all such insults, not just "nigger." The offense is "a class C misdemeanor," which may be punished by a fine of up to $500, the same fine the ordinance would impose.
UPDATE: Brazoria has abandoned this effort.