I think this latest plot arc moves to fast and there seem to be questions and inconsistencies with respect to Duke Xianguo's motivations. The question why seek a match between his daughter Manyi and Jinqi (who is in camp Crown Prince) when he means to support Prince Han? Upon rewatch, I think I have managed to sort most of it out although I still think there are some inconsistencies.

It is known that the emperor is ailing and even though he has named his successor, there should still be politicking at the highest levels while the emperor can still change his mind. This is a time where both the Crown Prince and Prince Han should remain outside the capital until summoned to return by imperial decree (meaning the emperor's mind is made up). This is to prevent either one of them pre-empting the decision - whoever is in the capital at the time of the emperor's passing is the most likely successor regardless of whether its by imperial decree.

Duke Ziangguo Zheng Baichuan has up till now been neutral - after all he is maternal cousin to both princes so whoever ascends the throne, he wins. No need to pick sides. But initially he clearly wished to be closer to the Cown Prince since he is already the named successor. But he finds that the CP is already surrounded by people like Marquis Yuan and Lin Jinqi who have earned favor from him via merit (i.e., their investigation of the military embezzlement case and defeating Wo pirates). At first he goes to great lengths to try to marry his daughter to Jinqi, who rebuffs her repeatedly. Father and daughter are vicious and vindictive people - they decide to take it out on Lanxiang, whom they see as their obstacle. This part is still thin to me - esp as far as Duke Xianguo is concerned as he has bigger fish to fry than try to go after a peasant girl who sings opera. But I can see Manyi taking her father's plan too far bc her nose is out of joint.

Meanwhile Duke Xianguo has decided to take Prince Han's side because the Crown Prince made himself vulnerable by coming to Jinling. He is the commander of the Jinling Joint Defense and in a unique position to detain the Crown Prince in Jinling, even make sure he never leaves Jinling alive. This would make him the person that puts Prince Han on the throne - the new emperor will owe him above everyone else. That is why as Jinqi and Marquis Yuan already figured out, he had no intention of allowing the CP to leave the Yuan mansion. Marquis Yuan will be guilty of failing to protect the CP and can be silenced before he can even defend himself. That is why he took such a big risk. He will be the kingmaker and in a position to even demand that Manyi be made empress. Whereas if the CP takes the throne, he will just be one of more than one noble family with a blood relationship with him - he won't be the one who put him there.

this is actually really helpful because I was really struggling with why he was so irrationally set on Jinqi because playing both sides with a permanent alliance didn't seem all that necessary since they already knew the emperor wasn't long for this world.

Indeed, the political aspect of the plot in this drama is badly articulated. While we've understood, more or less, the opening political conflict which led to the extermination of the Shen family, the script forgot to inform the audience what was going on on political stage. The writers only threw some crumbs from time to time here and there to let us get a vague idea of important power dealings which influenced the lives in Lin's residence. Among those "crumbs", we know there are corrupted officials who didn't only halt military provisions (sending cheap coats to soldiers and causing ML's demotion) but also secretely colluded with Wo pirates. The friendship between LJQ and Marquis Yuan was practically formed on their common goal to fight them (both the domestic corrupted power and the pirates).
Although we've never seen (on the screen) the Zhengs as a part of that domestic corruption clique, we have some elements to assume prince of Han (CP's bro) relied exactly on that force (we've heard what duke Zheng was screaming in prison) as a base and in order to take the throne from his bro CP. Prince Han will remain unpunished for the time being (as it happened to Zhu Di's younger son in history, who rebelled later on, after his brother died as an emperor).
The question is how the people in Lin family (espc. the patriarch) are so uninformed about what's going on around them? They readily marry their daughter to the marquise and then search marriage with a family from the opposing party? Why ML doesn't tell anything to his grandfather or his mother, at least AFTER the evident rebellion of the Zhengs? Hey, you wanted me to marry a girl whose father was about to kill the Crown Prince! It doesn't make sense, that's why the scriptwriters prefered to undertone the entire political aspect, hoping people wouldn't notice or wouldn't care much about it. Other than beating HIM for choosing a wife, it is his airheaded parents and grandparents who deserve a beating before ancestors' tablets