First thing I'd do is a compression test both dry and wet. I'm leaning towards a bad set of rings or worn cylinder. And I'm wondering if a piece of ring may have busted off clipping your exhaust valve. In order for something big enough to get out of your air box thru the carb and intake tract and intake valve and damage your exhaust valve it would most likely get hung up causing damage somewhere else. If your compression test shows good for both tests I'd reverify your timing. A new plug is definitely in order as well. If all that checks out I'd reset the carb jetting to factory as it may have been tweeked to run without the air filter.

 

Check that the tank vent isn't blocked first, but problems with engines that cut out when hot are usually spark related, so a new plug may cure it. Next check the charge rate, although supposedly totally separate from the ignition, I find an overcharging (over 15v) regulator can affect running. However it is usually a stator problem, as this gets hot quickest. I had an Ozark which would cut dead after about 10 min. Got the stator tested by a specialist who said it was OK, but managed to check the power and trigger coil voltages when the fault was on and found no power from the trigger coil, fitted a new one and the fault never recurred.