The audience is supposed to assume that the fictional Klingon characters continue to speak Klingonese for the entire duration of those scenes, but the movie production shifts from using subtitles to interpret their dialog to dubbing in English language dialog in the midst of those scenes.
You can assume that the fictional Klingon characters actually speak Klingon all the time in every scene except when talking to non Klingons when they might have to use a non Klingon common language, so it is an artistic (or possibly financial) choice by the director whether the actors portraying the Klingon characters speak Klingon or English in any particular sentence.
So possibly the director decided to start a scene with a few lines of the Klingon characters speaking Klingon to each other with subtitles in English, and then switch to having the Klingon characters speak English for the rest of the scene in order that English-speaking audiences don't have to read subtitles fast for the entire scene.
Possibly the Human actors portraying the Klingon characters do multiple takes of every shot, sometimes speaking their lines in Klingonese and sometimes in English. Later in post production two versions of the scene were assembled, one with every line in Klingonese and one with every line in English. So the director, Leonard Nimoy, could choose which combination of Klingon lines and English lines to use in that scene by combining shots from both language versions of the scene.
Or possibly the director and screen writers had already decided with lines would be in Klingonese and which in English in the completed film and they only filmed lines of dialogue in one language.
Note that when characters say things in languages foreign to the audience of the film, there are three options:
One, leave the foreign dialog untranslated expected the audience to get a general understanding of the meaning from the action.
Two, use subtitles to translate foreign dialog.
Three, dub in translated dialog spoken by the original language actors or by other actors who speak the language being translated into better.
So when Klingon characters in Star Trek III: The Search for Spock seem to switch from speaking Klingonese to speaking English in the middle of a scene, one can assume that "really" only the Human actors switch from speaking Klingonese to speaking English, and that the Klingon characters continue to speak Klingonese, but the movie production switches from using English subtitles to using English language dubbing.