Corporate turboprop pilot reported a NMAC with a small aircraft while departing FPR airport.

Narrative

I was cleared for take-off on Runway 10R from FPR and told to climb on runway heading to 2000 ft. on an IFR flight. While leveling off at 2000 ft. I contacted Palm Beach departure as instructed and was told to make a left turn to a heading. I initially started the left turn while the ATC Controller was still talking and immediately saw a single engine; low wing aircraft just in front and to the left of my aircraft; closing extremely fast; at the same altitude; while receiving a TA warning. I was flying at 160 knots on the autopilot. I Immediately disengaged the autopilot and made a rapid emergency vertical climb with maximum power (way over torqued my engines) to the right to avoid hitting the other aircraft. We missed the other aircraft by less than 50 feet off as it went under my left wing. My airspeed dropped from 160 knots to less than 120 knots during this emergency climb.I believe my initial answer to the Controller while I was making this emergency maneuver when told to turn left was; 'I can't' (I had my hands full!!) the Controller once again told me to make a left turn and all I could say was 'no'.If I had made the left turn as instructed; we would have hit the other plane. I had 9 passengers on board and I disregarded the Controller's instructions to save my life and lives of my passengers.I would like to add that the FPR airspace was overloaded. The Tower Controller told multiple aircraft to stay out of the Class D airspace because they could not handle the traffic at that time while I was waiting for my take off clearance.Fortunately it was a near miss and not a mid-air fatal accident.