Word to drink something quickly, as in 'down' or 'chug' a shot

quaff

: to drink a large amount of (something) quickly.

M-W

He quaffed the shot in one gulp, grimaced, and ordered another.

Moonlight in Vermont: A Novel

Describing the physical action required as opposed to the act of consummation conjures an effective image. When consuming a shot glass of alcohol quickly you are not sipping or even slurping it; you put it to your lips and throw your head back to splash the liquid into your mouth as quickly as possible.

He threw back a shot of whisky.

I don't know if that is within your scope of 'formal' but on the other hand, people aren't often 'throwing back a shot' at formal occasions.

ingurgitate:

to drink largely, to swig (A Dictionary of the English Language, By Samuel Johnson and John Walker; accessed via Google Books)

Merriam-Webster:

to swallow greedily or in large quantities [...]

Did You Know?
Most people are familiar with "regurgitate" as a fancy synonym for "throw up," but far fewer know of its rarer antonym "ingurgitate." It's a word as likely to turn up in a spelling bee as in a conversation, but it does see occasional use, both literal (as in "ingurgitating red wine") and figurative (as in "ingurgitating artwork"). "Regurgitate" and "ingurgitate" (as well as "gurgitate," an even rarer synonym of "ingurgitate," and gorge, meaning "to eat greedily") can be ultimately traced back to the Latin word for "whirlpool," which is "gurges."