I agree the brother needs to tone it way down, but that's not the only thing which he's failing to consider.
They're on a different world, and he's assuming that everything works the same way it does at home, that there's an aristocratic hierarchy, and peasants are peasants. Refuses to listen to his own kin, who is clearly more experienced with the people, and refuses to stop causing trouble. Being willing to show off his strength every single opportunity he gets to brute force his way through. So he deserves every ounce of what Christina is dishing out for being so bullheaded.
The mage, once again I point out the kind of world they come from. Our common sense doesn't apply to their way of life or thinking, so like Christina's brother, some adjustment is necessary. Bartholomeus is a Knight, and his position commands a certain amount of influence and respect where they're from, and if the mage considers herself his fiance, to their way of life, she also considers herself subordinate to him, particularly since he and Christina come from a noble house. His mistake is obviously that he thinks his name, and the name of the kingdom they serve actually means a damn on Earth. They don't live on 21st century Earth even if we do, and we can be glad that we do, and that this story is fictional.
For Kanji keeping a lid on the incident, remember how Christina was treated when they found out about her. And they had her full cooperation, so it was tedious, but for the most part she was treated well. Now we have her obviously volatile brother who doesn't hesitate to resort to violence. He's lucky Honami doesn't have armed JSDF Marines out there with her. Any one of them will probably shoot them dead before Bartholomeus or Louise can finish any incantations. Any further hostilities, and the two will likely be encarcerated, and treated like lab rats for the rest of their natural lives, then have their individual body parts vivisected and sent off for study all across the planet. So Barholomeus needs to get it in his head fast, that even at home, for what he's doing, a foreign national in someone else's country resorting to violence can easily be treated as an enemy combatant.
edit: Kanji is doing this for his wife's sake so she won't have to watch her brother treated like a criminal. At the least, he's doing his earnest best to get along with his brother-in-law instead of throwing him under a bus, who's too hung up on class differences to care about how well they've done for themselves, or that his sister attained her happiness without his supervision or input.