Balancing insurgencies is probably the toughest part of the mission making process. What I like to do is limit the friendlies to larger objectives only (so medium and large in a lot of cases), and sometimes only allow them to be aware of Military objectives (so your commander would only have a Military Placement Military Objective module) so the humans can handle all the Civilian Objectives without any interference.
Then once you clear the area, to hold it, use the Commander Tablet to call in reinforcements to garrison it with either the Logistics menu or rerouting friendlies in the Operations menu (clearing their waypoints and adding new ones).
In my experience the Asymmetric Commander takes a more defensive role in the beginning hours, while he establishes his Insurgent HQ’s and recruits civs to join his side. To me this makes sense that it would be his priority first and foremost. Your job would be to stop it.
As his forces grow, you should see these new insurgent groups get sent out to objectives. Regarding your friendlies staying in their areas, make sure their Commander module is synced to the Insurgent Objective Placement modules if you want them to take a more offensive approach.