Neither is quite right, but both are close, and they have different meanings. "Furthermore, that video shows THE disdain OF his ministers for other institutions." OR "Furthermore, that video shows his ministers' disdain for other institutions." Here, "disdain" is treated as an emotion, like love or hatred. "The flour and sugar on the pantry shelves show the love of the chef for cake"= "...show the chef's love of cake." We need to add "the" because the prepositional phrases at the end of the sentence specify WHICH disdain we are talking about. “ Furthermore, that video shows disdain BY some of his ministers TOWARDS/AGAINST other institutions." Here, "disdain" is shorthand for "actions caused by the disdain", just like "criminal neglect" means "action or inaction that results from a neglectful attitude." "The prosecution will demonstrate criminal neglect (committed) BY the defendant AGAINST his children on at least three separate occasions."

I think 'of' sounds better here. I'm not 100% sure but I think nouns are followed by 'of' and verbs are followed by 'by'. 'Video shows disdain of someone for something' 'Video is presented by someone.' Hope it helped 😊

of