ok so the reference went way over everyone's head. It's ok, it's a Japanese meme.
Teiou is imitating Rudolf's face and twisting her catchphrase that went viral in JP social media from the original manga.
Rudolf's final line is not cringing at the joke but at how it references a rather intimidating Rudolf. As a student council president, it is not something that Rudolf wants to be remembered for.
To make matters even worse, this is technically a child imitating a hostile quote by her father.
Honeyy! Honeyy!
Ho-ne-yyyy!♪
I've bought some for you, Teio.Woweee!
Thanks, President!
Don't lick down on honey.
The word "to look down"/"to make light of" in Japanese is pronounced the same as "to lick", both /nameru/.Ugh... Teio...I'd appreciate it if you could at least not turn that line into a joke...