WW24 FLT CREW AT 3000 FT WITH SCT; EXPERIENCED TCASII RA DURING VISUAL APCH VECTORS TO CRQ.

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WE WERE ON AN IFR FLT PLAN; VECTORS FOR A VISUAL APCH RWY 24 CRQ. WE WERE STILL WITH SOCAL APCH ON A HDG OF 060 DEGS; EXTENDED DOWNWIND FOR LNDG SEQUENCE AT 3000 FT. OUR TCASII SHOWED TFC AT OUR 1:30 O'CLOCK POS AND ABOUT 2 MI; 200 FT BELOW US. ALSO; ANOTHER TARGET AT OUR 12 O'CLOCK POS APPROX 4 MI AND 300 FT BELOW US. APCH CTL TOLD US TO TURN R TO A HDG OF APPROX 150 DEGS FOR R BASE. WE ADVISED HIM OF THE TFC ON OUR TCASII AND THAT HE WANTED TO REMAIN ON OUR EBOUND HDG AND CLB TO 3500 FT TO AVOID TFC CONFLICT. HE SAID OK; THEN WE RECEIVED AN RA TO 'CLB.' AT THE SAME TIME; THE CTLR ASKED US WHY WE COULDN'T COMPLY WITH HIS REQUEST. WE TOLD HIM ABOUT THE TARGET HE WAS GOING TO TURN US INTO. HE SAID HE DIDN'T SEE ANY TARGET NEAR US ON HIS RADAR. AT THE SAME TIME WE STARTED OUR CLB; MY COPLT SPOTTED THE TFC VISUALLY; JUST WHERE THE TCASII SAID IT WAS. AFTER WE STARTED OUR CLB; THE OTHER ACFT ALSO STARTED A CLB. I CLBED TO 3700 FT AND THE TFC CROSSED UNDER US AT ABOUT 300 FT BELOW US. I DO NOT UNDERSTAND WHY HIS RADAR DIDN'T SHOW HIM THE SAME 2 TARGETS OUR TCASII SHOWED US. WE SAW THE 1 TARGET VISUALLY; MY COPLT DID; I WAS FOCUSED ON CLBING OUR ACFT RAPIDLY AND LOOKING AT THE TCASII TO MAKE SURE WE WOULD CLR AND STAY ABOVE THAT AND THE OTHER ACFT. THE TCASII UNIT IN OUR ACFT IS A $180000 UNIT; ONE OF THE BEST AVAILABLE FOR CPR ACFT.