ACR HAS CONFLICT WITH TFC SIGHTED THEN LOST.

Narrative

WE WERE FLYING THE HOMEY ARR INTO PBI. APPROX 40 MI NW OF PHK; ATC CLRED US DIRECT PHK DIRECT PBI AND CALLED VFR TFC AT 2 O'CLOCK LOW. I LOOKED DOWN AND SAW A MLG ABOUT 10000 FT BELOW US. I ANSWERED THAT I HAD THE TFC INSIGHT; BUT I THOUGHT HE CLRED US DIRECT HOMEY. HE RETURNED WITH 'CLRED DIRECT PAHOKEE DIRECT W PALM; CROSS 15 MI N OF PAHOKEE AT 15000 TFC PERMITTING.' I STILL THOUGHT HE WAS CLRING US DIRECT HOMEY; SO I READBACK THE CLRNC ONLY SAYING DIRECT HOMEY. HE ANSWERED 'READBACK CORRECT; EXCEPT CLRED DIRECT PAHOKEE.' FINALLY UNDERSTOOD; BUT IN THE PROCESS LOST THE TFC. WE CONTINUED OUR DSCNT. AT 17000; APPROX 24 DME NW OF PHK; THE TCASII CAME UP WITH A 'CLB' RA AND COMMENCED A 2000 FPM CLB. THE VSI SHOWED A RED TARGET 1500 FT AWAY. WE CLBED IN COMPLIANCE WITH THE RA AND ADVISED ATC. THE 'CLB' RA CONTINUED AND THE TFC GOT AS CLOSE AS 1000 FT INDICATED ON THE VSI TARGET. WE THEN SAW THE MLG OUT THE L WINDOW AT OUR ALT. CTR SAID THAT HE WAS IN A VFR CLB. HE FINALLY SAW US AND BANKED HARD TO THE L. THE TCASII GAVE US THE 'CLR OF CONFLICT' MESSAGE SHORTLY THEREAFTER. I WAS VERY SURPRISED TO SEE AN ACFT DOING A VFR CLB IN SUCH A BUSY FUNNEL OF AIRSPACE; BUT I SHOULDN'T HAVE LET MYSELF GET SO WRAPPED UP IN THE RADIOS TO THE POINT THAT I LOST SIGHT OF THE TFC. I WAS ALSO SURPRISED THAT THE CTLR NEVER SAID ANYTHING TO US WHILE HE WATCHED OUR TARGETS MERGE. TCASII WORKS. IN THIS SITUATION; IT COULD HAVE VERY WELL SAVED US AND OUR PAX.