Is it permissible for my wife to undergo a Caesarean section, with our prior knowledge that her 'awrah will be exposed to non-mahram men, due to the unavailability of a full female medical team?

The unavailability of a complete female medical team for a C-section delivery does not, Islamically, prevent a woman from seeking pregnancy and childbirth, even if it entails male doctors examining her. The duty is to strive to provide a female team, but if it is not available, then Allah burdens not a soul beyond its capacity.