Unfortunately I couldn't watch the clip -- your link took me to a playlist rather than a specific episode.
No, they're not synonyms.
"mettle" isn't a common word and is only really encountered in the expression "to test one's mettle" where, as you know, it means to see/interrogate what someone is made of.
"nettle," by contrast, is a very common word - we have a lot of this plant in the UK - and "to grasp the nettle" means to engage wholeheartedly with an uncomfortable or difficult situation.
Therefore, someone's mettle can be tested by the alacrity with which they grasp the nettle but the similarity of these words ends with how they look and sound. Using them together, the writers are playing with the sounds of the language for comic effect but it doesn't reveal synonymy. It's fun wordplay though!