If you reckon that Blue Lake is a moderate approach then you might want to consider the Australian Rogaining Championships at Providence Portal (near Adaminaby) - it would be a doddle!!
Black Range bouldering would be perfect.
Mt Corree is significantly cooler than Canberra but can be quite sunny. Online guide used to be available from Canberraclimbing.com. Otherwise you could just head for Jerusalem (as per ACT Granite) and try everything around to the right of it. From memory the classic routes from Jerusalem are 23 (laidback international - the lay back flake start), 22 (superjesus - underecut start then crimpy arete/face), 24-26 (on the big steeper wall) and then a 22 further round. Most of the other routes in the area are easier than 19. The drive up can be brutal on a low clearance car. Consider parking half-way up when the going gets bad.
Orroral Ridge might be an option for a bit of shade and an easy trundle along the ridge from the carpark.
Even Booroomba can be significantly cooler (5C) than Canberra. Around the South Buttress there are usually shaded areas esp in the morning.
As for camping, Honeysuckle campground is pretty good. Near Mt Corree you might just have to bush camp in the semi-closed campground at the base of the drive up the hill (about 5 km from the crag). Really nice spot before the fires, I am not entirely sure that it has been reopened for fear of falling timber.