I forgot that the Kzinti also had a mention in The Infinite Vulcan so that's one appearance and one mention for them versus one for the Organians and one mention of a treaty with their name on it.
As far as fitting into the Trek Universe let's see what D.C Fontana and Larry Niven have to say about it, shall we?
From Memory Alpha:
Following D.C. Fontana's invite to write for the animated Star Trek, Larry Niven proposed two plot ideas that went undeveloped, as both had content that was deemed to be too problematic for the series. Shortly thereafter, Niven and Fontana were spending part of an afternoon at Gene Roddenberry's residence when Roddenberry suggested that Niven rewrite "The Soft Weapon" as an episode. (Playgrounds of the Mind) Fontana reflected, "We told Larry, 'Just write it, but put in our characters. You can put in the Kzin and all that.'" (Star Trek: The Magazine Volume 1, Issue 16, p. 67) In retrospect, Niven remarked of the adaptation idea, "That worked." However, he was initially not entirely sold on the prospect of including in the episode the Kzinti, who feature in the original short story as well as elsewhere in the Known Space universe. "I thought hard before giving the Kzinti to the Star Trek universe," Niven acknowledged. "I did it because I thought it would be fun to see what others would do with them."