RPTR CLAIMS NEAR MISS WITH LIGHT TWIN AFTER HAVING THE TFC IN SIGHT AND BEING ISSUED A CLB CLRNC WITH THE OTHER ACFT MAINTAINING VISUAL SEPARATION.

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ON DEP FROM FT MYERS REGIONAL SOUTHWEST ARPT (RSW); RWY 24; WE WERE GIVEN INITIALLY A CLB TO 5000 FT AND A L TURN TO 060 DEGS. DEP CTL ADVISED US OF TFC AT OUR 2 O'CLOCK POS AND 6000 FT NBOUND. WE VISUALLY PICKED UP THE TFC AT OUR 12 O'CLOCK POS APPROX 2-3 MI AHEAD. DEP CTL ADVISED THE ACFT (ACFT Y) OF OUR POS; ASKED IF HE SAW US; AND WHEN HE SAID HE DID; TOLD HIM TO MAINTAIN VISUAL SEPARATION FROM US. HE SAID HE WOULD. DEP THEN TOLD US TO CLB TO 10000 FT. IT BECAME OBVIOUS PRIOR TO THE TCASII ALERT THAT WE WERE ON A COLLISION COURSE IF WE CONTINUED OUR CLB RATE. WHEN THE SEPARATION WAS LESS THAN 1000 FT; WE LEVELED OFF; MISSING THE ACFT BY 400 FT (TCASII). I INFORMED DEP CTL OF THE NEAR MISS. ACFT Y DEFENDED HIS ACTIONS (IF THERE WERE ANY). THIS HAPPENS ALL TOO OFTEN; WHERE ATC TRIES TO AVOID CTLING ACFT AND DEPENDS TOO HEAVILY ON VISUAL SEPARATION. LOOKING BACK; WE SHOULD HAVE LEVELED AT 5000 FT UNTIL CLR AND NOT RELIED ON THE VISUAL SEPARATION. DEP SHOULD HAVE PROVIDED THE SEPARATION.