Corporate jet pilot reported receiving a low altitude alert from ATC; during a maintenance flight in daylight visual conditions; on initial climb from a tower-controlled airport. The pilot indicated a late climb while being distracted with high workload; maintained terrain clearances; then resumed the climb and continued the flight.

Narrative

On departure out of VNY; on the Canoga3 departure; we missed the climb gradient required by the SID. This was a maintenance flight; so we were already working in an abnormal situation. Because it was a maintenance flight; we didn't carry that much fuel; and we accelerated and climbed very quickly. On takeoff; we believed that the autothrottles were armed. While trying to comply with the crossing restriction in the NOTAM for the Canoga3; we received an overspeed warning for our flap configuration. Usually the autothrottles will pull back to protect the aircraft. After a quick instrument scan we realize the autothrottles were no longer armed; so we had to manage that situation. While managing the overspeed; trying to be cognizant of the crossing restriction; and flying the lateral confines of the SID; the controller issued a low altitude alert and then a traffic alert. We were task saturated and missed the point where we needed to start climbing. We did indeed start climbing; we were just delayed as we managed all the tasks at hand. The controller rightfully called us out for it; and handed us off to the next controller. We were VMC and could maintain terrain and traffic clearances at all times. We did Not receive a TCAS alert or ground proximity alert from our avionics.