You were testing the car in pretty extreme conditions. If you had the A/C on full and you were actually cold, it requires a tremendous amount of energy to keep a car cool when its over 100* outside and the sun is shining in the car. Your compressor was probably going full blast to keep up with the demand and your condenser was throwing 130*+ air at the radiator and you expect it to keep the coolent at 180* (only 50 degrees differance).
I will say that 240* is way to hot and would worry me too, but you still had a very extreme situation going on. I'd check to make sure your getting good air flow through the radiator (like chacking if the front air dam is in place and the plastc belly pan under the front of the car is still there) Also, make sure its clean. I'd try cleaning the radiator and condeser with Simple Green to remove and grease or grime that'* robbing your car of cooling power. Don't use a pressure washer, you'll bend the fins. Only use a low pressure garden hose, and then on a misting setting, no streams.
As for a flush, you won't actually gain much cooling performance. Flushing keeps the additives in the coolent fresh and keeps the coolent from corroding the inside of your engine. It'* still a good idea, but probably won't fix your problem.