Said Knight's family was captured by a robber baron with an axe to grind. The villain wants to ransom the family for a hefty fee. The PK chooses to squeeze his manors. He has some famous traits (16+) that are the opposite of the traits checked by squeezing. Must the knight roll successfully on the "squeeze" traits to be able to force himself to squeeze the manors? And if he's doesn't successfully roll these, does that mean he cannot force himself to squeeze the peasants? Let's see if I can squeeze some answers from you all. :)

I would make him roll them. The point of famous traits is they take away agency in exchange for bonuses. The player always has the choice to act against the trait and check the opposite which will generally be a loss come Winter.

I'd have him roll first which trait would get triggered (in addition to Arbitrary). If he has 16+ in either opposite trait, yes, I would make him roll in order to do it. If the player goes against his 16+ trait regardless of the roll, then I would lower his Famous Trait automatically, in addition to giving the Squeeze trait check. Yes, it is a 'punishment' for violating the 'famous trait dictates your actions'. Heck, I might even go as far as to tell the player that no, his character would not do that. As much as I am for the PK being the player's property, there is an explicit agreement that if he has a Famous Trait, his character strongly acts like it.

I might have his Famous Traits be rolled opposed by his Love of Family, especially if that Passion is Famous as well. If Love wins, he's will to take the check, if the Trait wins, he can't bring himself to act that way, even to help his family. Since he's not trying for Inspiration, I would not allow him to go Mad, though he may become Melancholy depending on the circumstances if he fails.

Agreed, the PK can use his passions to contest his principles. I wouldn't make it result in madness either, unless his inaction causes the death of his family. Then a period of madness would be appropriate.

I might require a single opposed roll of Love (Family) versus the highest trait in question, just to minimize die rolls, but Deacon's solution seems otherwise sensible.

I agree with the majority here. Notable passions give glory. These same make you famous. They are your default characteristics. Going against them means there must be a penalty, else everyone would be going against their passions whenever the situation becomes uncomfortable. I allow successful rolls of passions to affect whatever the PK wishes as long as it fits the story and the character. Deacon has a good point and is how I would handle it.

Passions and traits of 16+ mean you are notable for them and going against them means you get a punishment.
I would also make him roll the traits opposed to his Love(Family). Whichever wins means you act accordingly. If you still go against them then checks and/or reductions are in order. So if the famous traits win and you cannot get yourself to squeeze his peasants means you lower your Love(Family). If the love wins you get checks to the opposed traits, which means the famous traits will be lower next year.

I always give the player the option to go against what is rolled, but it will have huge impact on his traits. Probably the reduction of famous traits as a consequence.
I always remind my players that the game sometimes does not have good or bad choices. It is a game of consequences and as such each choice you make has a downside. Are you willing to take it?

There are of course other options to the PK, like asking his Lord for help or his friends. He may even turn his attention to a moneylender, but of course all these options have consequences as well.