You see his reaction in TSFS, referring to them as "Klingon bastards," instead of "bastards," which implies (strongly) using their species name as a pejorative (the way a white racist would say "n***** bastard" instead of the latter, which would have been enough for one without racist intent). Add his TOS behavior in "Errand of Mercy," "Friday's Child," and general tone in "A Private Little War," and he does not seem to be the biggest fan of Klingons, or even tries to imagine part of the race may not be like the military end he faced.
With David's murder, his undercurrent of hostility evolves into its full-on, racist form. His behavior in TFF was more about putting up with them until he could separate from them.
You and EliyahuQeoni have excellent points. I never understood why it would be inconceivable for Kirk to be prejudiced towards Klingons. He's not a perfect man, by any means. He's fought them his whole life and lost his own son to them. Of course it'd be great if he could see outside his own experience a bit sooner and lose his prejudice, but he's flawed, he's human. And I've always found humans more interesting than paragons.
(I also find Vulcans more interesting than paragons and thus would really have liked Saavik's story arch to end in TUC with her being the traitor. That would have had emotional impact.)