What's with that shot? I can't imagine there being any good way to redeem his brother after that kind of display.
I guess...we can take mom aside and treat it as doing the first empress a favor getting rid of an eyesore?
You've gotta remember, everyone wants to use the first prince as a means of protection/ power-grab. Getting involved with others in any shape or form sends a message to the royal court. The whole reason why they are even meeting in the first place in that public forum was because Ruldoph was finally a valueable asset. Him being cold shows others that he's harder to manipulate.
For context, remember how both Rudolph's first and second personal maid both used their close relationship with him in order to manipulate him (with Tappel using him for personal gain and Maya manipulating in order to benefit her true master, Lisbeth). The first prince is leagues more influencial than Rudolph so he's likely getting approached by several manipulative people a day. He can't afford to be more inviting/open otherwise he'd be taken advantage of.