We've stayed at Broken Arrow and Diamond Lake CG. Diamond Lake CG will take reservations for some of their spots. Broken Arrow is 1st come 1st served unless you're wanting a group site. Then you can reserve.
We have cruised thru the Diamond Lake RV park and it's a typical RV park. No privacy and too close together for us.
Any of those spots are great as a base camp. We've been there many times. There's some good hiking trails nearby as well as a 12 mile paved bike path that circles Diamond lake.
The mosquitos are a problem early in the season, but by mid-late august aren't much of a problem. We've been there in July and could barely get the trailer setup before we were inundated with them.
We've been there in August and saw hardly any. It helps to get a spot at Diamond Lake CG near the water. If there's some wind that keeps the skeetors away. You should be fine in Sept. It's well worth the trip. Some of the prettiest scenery around.