How would you call the pout that one makes when assessing a situation? Is there a word or phrase to designate this kind of grimace?
I think you would just call that a frown (note that it can indicate concentration as well as negative emotions like disapproval or displeasure):
A facial expression indicating disapproval, displeasure, or concentration, typically with the corners of the mouth turned down.
‘she leans back uneasily, her mouth forming a pensive frown’
‘a frown of disapproval’