Under what condition can the site commander approve lodging of officer and enlisted females together?

The correct condition under which the site commander can approve lodging of officer and enlisted females together is when operational needs dictate. This means that in situations where the mission requires flexibility or specific arrangements to maintain operational effectiveness, the site commander may authorize this lodging arrangement. These operational needs can arise from various factors such as the nature of the assignment, the urgency of deployment, or the requirements of the mission at hand.

The other choices reflect considerations that may not necessarily align with established military policies. For example, limited space can create logistical challenges but does not warrant a formal exception to gender lodging policies. Similarly, policies are typically set to maintain professionalism and operational integrity, and while training exercises may offer temporary exemptions, they do not provide a blanket approval for the lodging of mixed-gender units across all scenarios. Thus, the authority of the site commander to make exceptions aligns directly with operational needs rather than the other options presented.