Interesting note here (to me, anyway...) According to the NTSB report it appears that the aircraft was NOT overweight (although quite close to max gross.) Which leads me to some interesting thoughts.
First, given that he communicated to the tower that he was "a little overweight" - when he apparently was just within gross, according to the manufacturer's calculations in the NTSB report - makes me wonder whether he did a weight & balance calculation in the first place. And if he didn't, but (correctly) suspected he was close, I wonder if, when he found he had poor climb performance, he jumped to the wrong conclusion, and focused in on the "overweight" issue, possibly missing the flaps.
Another interesting point, if he had recognized the flaps issue - possibly even if he had noticed earlier in the takeoff - the NTSB report points out a significant difference between the SR20 flap actuator and the one in the 172, in that, in the Cirrus, you move the actuator to a detent command position that you want the flaps to go to, then you can turn your attention (and your hand) elsewhere while they transition. In the 172, the actuator is a spring-loaded "up/down" switch, and you have to actuate and hold the lever until the flaps actually reach the desired position.
I wonder if, having taken virtually all his training hours in the Cirrus, and likely being a low-time pilot in the Cessna according to the report, whether he might have reverted to instinct during a high-workload moment, and thought he commanded the flaps to retract by briefly flipping the Cessna actuator "full up", and then letting go of it (which is how you'd do it in the Cirrus.) However, in the Cessna, this would leave the flaps essentially fully down (if they were there to start with.)
Note: these comments are speculation - guesses. I mean no disrespect to the pilot, nor is this intended as a personal judgment or "piling on" - only trying to draw lessons about cockpit procedures, problem solving, and decision making from the example this provides.