Why do you think Cardan did not defend Jude more actively during the Queen of Mirth thing other than giving her some really vague warning? Even apart from his feelings for her, letting his advisor be humilated by gentry must make him look weak as a ruler. And we know that Locke did try to humiliate Cardan indeed. I mean, all Cardan had to do, was play "She is mine, you don't get to play with her" card like he did with Orlagh and send her to the palace but he chose to play along (I like Cardan but for this scene specifically I hold a grudge).
While we are on the topic, why do you think Taryn got zero consequences for helping to steal half of his army? I get that he probably had to keep it in secret in order not to reveal his deal with Jude and he couldn't be too cruel for Jude's sake but like he did nothing at all? Didn't even sent her away from the court? He didn't even have to reveal Taryn's part in the Madoc's scheme, his reputation is bad enough so he could just honestly say he hates her and forbids her to attend the court? Not to mention she is the sister to the exiled seneschal and the daughter of a trator? But again, nothing. Honestly I don't know how both Cardan and Jude got their bad reputation given how lenient they are sometimes.
Based on what the story shows about Cardan, I read a few different motives. I'm sure part of it was not wanting to give Locke any more reason to single Jude out for his games against Cardan, and maybe some of it was to not throw gas on the rumors about the nature of Jude and Cardan's relationship. The main reason, though, is probably because Jude needed to be at the revel.
He did try to warn her at the beginning of the party, and I actually think he did a good job. He pulled her aside to tell her that he made sure that Taryn wouldn't be there that night- a hint and a half that whatever was about to go down would be unpleasant for her. He told her she shouldn't be there. He even told her to go back to the palace, but don't forget, because of their agreement, he couldn't actually order her away. The choice to stay or go was ultimately Jude's, and Jude had her own reason to be at the revel that night. She was supposed to meet with someone from the Court of Termites. Even if she didn't want to be at the revel (and she didn't), she couldn't not show up. Cardan was probably aware of that. There was literally nothing he could have done to get her to leave if she was determined to stay, but he did try. Yes, his warning was vague, but that was most likely because he didn't know what was going to happen.
Cardan didn't know what Locke had planned. Not really. I'm sure he was aware that whatever it was Jude wouldn't like, but that's not saying all that much. He didn't know that Locke was going to make her Queen of Mirth. From what Jude says, it sound like it wasn't a thing that happened all that often. As for why he didn't put a stop to it when Locke named Jude the Queen, I can only guess. He wasn't happy about it. Cardan clearly seems pissed. Jude thought it was directed at her, and maybe he did want to punish her a bit for everything that happened between her crowning him and that moment.
What I think, though, is that a lot of it had to do with Cardan not really feeling like the king. I think he very much had imposter syndrome and fell back into his habit of mitigating harm done to Jude rather than throwing his weight behind stopping it. He was probably expecting her to refuse, and he would have let the issue die there. Or maybe it's because his brain stopped working when Jude took her dress off. The frustrating thing about First Person POV is that we only get the situation from Jude's perspective, and Jude is an unreliable narrator. I genuinely don't think that Cardan would have forced her to play along if she refused, though. He didn't have to dance with her. He could have let her get trapped dancing with the revelers for a while.
On the subject of why he let Taryn get away with what she did, Cardan probably left her alone for Jude's sake. Remember, Jude wasn't actually banished. Cardan expected her back any day, and he knew that she loved her sister. Of course he wasn't going to have Jude come back to find Taryn in the Tower of Forgetting. Now, had Jude decided that her sister should be punished for what she did, Cardan would have gone along with whatever Jude wanted, but Cardan wasn't going to do something that would upset his wife.
Also, Cardan probably didn't register exactly how big a deal what Taryn and Madoc did was. Madoc hadn't made any major moves against Cardan at that point. I doubt it was even on Cardan's radar at the time.