I guess many of us are naive about this issue - hopefully a discussion here will, at the very least, turn on some lightbulbs in our brain-housing-groups...
In this matter I strongly recommend that best approaches are through host nation government, local, and military figures rather than direct, regardless of relative depth in Islam and local culture. I would also say that this is not something that we turn to the Chaplains--who do great work regardless of faith--as they will be judged by their uniforms and their faiths. Indeed this is a subject of great debate within Chaplain circles as a possible conflict with their mission of religious support. The last thing I would ever recommend to a commander is to send military envoys/patrols into the mosques as an entree for engagement. I have been in mosques in Egypt, Syria, Jordan, and Turkey. All were famous and the mosques were used to seeing non-Muslim visitors. We were, however, always escorted.

Back to the indirect approach, the best way if you seek to engage the mosque leaders is to draw them into discussions outside the mosques and again that means with local contacts, tribal, government, security, etc.

Best

Tom