What was Klunk's motive for helping Ratchet defeat Courtney Gears in Ratchet & Clank 3: Up Your Arsenal?

In Ratchet & Clank 3: Up Your Arsenal, Clank gets replaced with 'Klunk' by Dr Nefarious, after filming an episode of Secret Agent Clank at the Holostar Studios. Ratchet is unaware of this, and continues his journey with Klunk none the wiser.

Eventually, Klunk reveals his true intentions (in Metropolis on planet Kerwan). Dr Nefarious explains that he replaced Clank, has a laugh, and teleports away, leaving Ratchet to battle with the transformed 'Giant Klunk' in order to rescue Clank.

But (literally) just before this happens (1-2 levels away), Ratchet and Klunk infiltrate Obani Draco, an artificial moon used by Dr Nefarious to experiment with the Biobliterator. There they confront Courtney Gears (who is in league with Nefarious), and do battle with her, eventually defeating her. Klunk then quips:

"One disposable Pop-star... disposed"

But why does Klunk not team up with her?

• Courtney knew who Klunk was, having been part of the plan to capture Clank in the first place. They could've teamed up together and taken Ratchet out, making the point of Klunk confronting him solo later a moot point.
• Courtney was a valuable asset to have, as far as Dr Nefarious was concerned. As a vocal supporter of Nefarious' plan, she could've been used as a front to promote the transformation as a good thing (and in fact was already doing so). As such I don't buy the 'disposable pop star' line. She was worth far more than was implied.
• Klunk was in communication with Dr Nefarious (this was shown throughout the other cutscenes). Even if he wasn't sure whether he should reveal himself at that stage, he could've asked Nefarious for guidance.
• At the time, Dr Nefarious was only really concerned with getting revenge against Qwark, and didn't see Ratchet as much of a threat. He idolised Clank (for his role in Secret Agent Clank), and that's what drove him to offer Clank to join his side in the first place. But he made Klunk to spy on the Q-Force in general, and keep tabs on their movements, as opposed to Ratchet specifically. Getting him to blow his cover, take Ratchet out, and fly back to the Starship Phoenix to continue spying on the Q-Force (whilst 'mourning') has no real downside from Nefarious' point of view.

I just re-watched all the game's cinematics again. Nowhere in the content of the game is your question explicitly answered, but there are a few possible (entirely speculative) explanations:

• Nefarious doesn't seem to like/care for Courtney Gears
• Nefarious is not prone to killing his opponents early as he seems to like to toy with them. As Lawrence clearly points out on many occasions, Nefarious is not very good at planning - likely written this way for comedic effect.
• Nefarious usually likes to deal with his adversaries himself. Especially given the personal history he has with Qwark.
• With the Biobliterator(s) in his possession, he doesn't really need to incite robot insurrection so Courtney Gears is rather disposable to Nefarious' plans. She's mostly a red herring to buy time to set a trap for Qwark and get the Biobliterator(s) operational. Every robot converted from a squishy would already be under Nefarious' control so it's more a matter of subjugating the existing robot population with the army the Biobliterator(s) would have gained him.
• Courtney Gears knew too much and this made her a liability. Nefarious doesn't truly trust anyone and, despite his offer to Clank, certainly wouldn't share his reign with her or anyone. The risks of her turning on Nefarious or revealing too much information were too high.
• As you said, Nefarious was unconcerned with Ratchet at that point, focusing all attention on Qwark, and since Ratchet was the only member of the Q-Force taking action, leaving Ratchet alive to draw Qwark into his trap was probably part of that decision. If Ratchet were killed at that point in the narrative, there was really nobody to give Qwark the confidence to go out on the frontlines himself. It's clear that Qwark is afraid to face Nefarious himself.
• It's also unclear how autonomous Klunk was so that may have been too far a deviation from his programming for him to take that initiative.