How would you react to an E2 who "smart mouths" you in formation?

200 pushups, humiliation in front of other trainees

Hand him or her over to a very capable and motivated NCO for some extra attention and go from there...

I have given you a lawful order" and you are you refusing to obey it, What is your reason for doing so? If not a good reason Take legal action at once. If in active combat any action needed depending on circumstance; such as costing the lives of fellow troops. it could include if needed a bullet for cowardice in the face of the enemy.

Something like this undermines your authority and rank and ability to lead and command which is why it can not be tolerated. Punishment needs to be quick and immediate and obvious for other subordinates to see so that they learn and realize that something like this is not and can not be tolerated in the military.

Automatic discipline! Severe verbal beat down of the bug in front of the others front and center. Double PT until I feel my point has gotten across.
If in boot camp then drop for 50 holding 50th at 1/2" off the deck until I tell you get your ass up and fall in. If this does no good then go up the chain of command.

Well in the Corps, at least 3rd Trax, LSB, LAR, Tanks, Heavy Weapons, 5th Marines, the 1st MAR DIV as a whole. My TOP or 1stSgt would have called me into the office and suggested that I visit the Marine in question in his room during Field Day. Or if an NCO or boot louie made the suggestion I would wait for a fueld op. Non-NCO's must maintain the discipline of Non-NCO's. Semper Fidelis my military cousins.

Article 15 for the young punk.

I think pulling him aside privately and then question him as to his motive. If has no reason other than laziness, I would tell him that he was restricted to barracks along with the platoon. If he didnt want to be a team member, then he can let them know it and then, if he still refused, I would send him to counciling under guard and have him evaluated for an ARt15.

I would slap the piss out of him. If that didn't work, we would go out back

Tell them to do it or they can clean out the crapper.