ma gavte la nata

But sometimes Belbo, when he became really angry, lost his composure. Since loss of composure was the one thing he could not tolerate in others, his own was wholly internal - and regional. He would purse his lips, raise his eyes, then look down, tilt his head to the left, and say in a soft voice: "Ma gavte la nata." For anyone who didn't know that Piedmontese expression, he would occasionally explain: "Ma gavte la nata. Take out the cork." You say it to one who is full of himself, the idea being that what causes him to swell and strut is the pressure of a cork stuck in his behind. Remove it, and phsssssh, he returns to the human condition.

I'd love if a native Italian speaker could tell me how much dialect ma gavte la nata is vs. pure Italian.

In standard Italian it would be "Ma levati il tappo" o "Ma cavati il turacciolo". Your average Italian would not understand the expression, because there is no word in Standard Italian akin to nata (on the other hand the relationship between gavte and cavati is quite obvious).
This being Umberto Eco's work, I would suspect the dialect to be from Alessandria. The people from Piedmont (Piemonte in Italian) are famous for their sarcastic wit.