I know countenance means appearance, behavior, and composure. What does the phrase 'to lose countenance, with a good countenance' mean?
OED: s.v. countenance n.1
6.a. ‘Calmness of look, composure of face’; ‘confidence of mien’ (Johnson); esp. in phr. to lose countenance, with a good countenance.
Essentially, then, “to lose one’s cool,” and “while maintaining composure,” respectively, and in both cases as expressed in one’s face.