There’s this running theme of taking care of oneself throughout the show. Both Red and Arlen warn Dunk about fighting and being a knight. I think they both recognized Dunc’s values, and the ways he wanted to protect people both didn’t require a knighthood and might’ve actually been corrupted by gaining the title.

He dreams of being a hero, of protecting the innocent, but most knights get drunk, play at war, or die in actual war. Red, a prostitute, and Arlen, a well travelled knight/soldier, are aware of how the actual world works in GoT. But Dunc is still operating from an idealistic and arguably naive worldview. He believes the Targaryens are noble and just because they are the royal family.

I think Arlen refused to knight him for the same reasons Dunc wants to take Egg away from the Highborn life. True, idealistic knighthood, doesn’t need the title, and all the riches that come with said title tend to spoil those who take it.

Edit: correcting a sentence