CL65 FLT CREW CLBS OVER VISUALLY ACQUIRED TFC CONTRARY TO TCASII RA TO DSND; ON DEP FROM LFT.

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AFTER DEPARTING LFT; WE WERE TOLD TO LEVEL OFF AT 2000 FT AND TURN R TO A HDG OF 290 DEGS. ONCE THERE; WERE TOLD TO TURN R TO 360 DEGS AND LOOK FOR TFC AT OUR 2 O'CLOCK POS. THE TFC WAS SPOTTED AND WE WERE GIVEN CLRNC TO MAINTAIN VISUAL SEPARATION AND CLB TO 10000 FT. WITH THE TFC AT 2 O'CLOCK POS AND 5 MI; WE GOT A TA. WE WERE CLBING AT 2500 FPM AND THEN GOT AN RA TO 'DSND.' THE TFC WAS AT OUR 2 O'CLOCK POS MOVING TO 3 O'CLOCK POS AND PASSING BELOW US AT ABOUT 3 MI AWAY. WE CONTINUED TO CLB; AS DSNDING WOULD HAVE PUT BOTH ACFT IN JEOPARDY. NOW WITH THE ACFT DEFINITELY BELOW US; WE GOT THE RA OF 'DSND; DSND' AND A COMMANDED VSI OF 4000 FPM DOWN. THE PITCH ATTITUDE AND CLB RATE WERE INCREASED TO CANCEL THE ADVISORY. THE ADVISORY STOPPED AND WE CONTINUED WITH A NORMAL CLB. WE WERE NEVER QUERIED BY ATC. IN THE FUTURE; WHEN 2 ACFT WITH SUCH DIFFERENT PERFORMANCE CAPABILITIES ARE IN CLOSE PROX TO EACH OTHER; ATC SHOULD WAIT TO ISSUE ANY FURTHER VERT CLRNC. WE SHOULD HAVE BEEN LEFT ON A HDG OF 290 DEGS; AS THE TFC WOULD STILL HAVE BEEN IN SIGHT; BUT WOULD NOT HAVE PRODUCED AN ALERT.