B737-700 CREW AND A BEECH JET HAD LESS THAN COMFORTABLE SEPARATION IN ZHU CLASS E AIRSPACE.

Narrative

TCASII RA GENERATED HDG DEV. CLRED BY HOU TO PROCEED TO AUS BERGSTROM AND DSND TO 6000 FT. DURING DSCNT; DIRECTED TO TURN L 20 DEGS AND EXPEDITE DSCNT THROUGH 16000 FT FOR IFR TFC AT 17000 FT. FURTHER CLRNC WAS DIRECT TO THE ARPT PASSING 16000 FT. AT APPROX 16500 FT; WE RECEIVED A TCASII TA WITH TFC 1700 FT BELOW AND AT APPROX 11 O'CLOCK POS. DSCNT RATE AT THIS TIME WAS APPROX 3000 FPM AND AIRSPD APPROX 300 KIAS. AT APPROX 16000 FT; WE GOT A TCASII RA 'CLB; CLB.' FO HAD JUST ACQUIRED THE TARGET VISUALLY AT APPROX 10-11 O'CLOCK POS. I HAD ONLY THE TCASII VISIBILITY INFO AT THIS TIME 10-11 O'CLOCK POS AND -600 FT. FO DETERMINED THAT A CLB WOULD ONLY RESULT IN STOPPING AT THE OTHER ACFT'S ALT AT BEST AND INCREASED THE DSCNT RATE IN CONJUNCTION WITH THE PREVIOUSLY CLRED R TURN TOWARD THE ARPT. I CONCURRED VERBALLY BASED ON MY ANALYSIS OF SIT FROM TCASII INFO. AFTER THE CONFLICT WAS RESOLVED; I ADVISED ZHU AND SUGGESTED THAT THE 20 DEG L TURN HAD REALLY INCREASED THE CONFLICT POTENTIAL; AND WAS THEN ADVISED THAT THE ACFT WE HAD MANEUVERED FOR WAS NOT THE ORIGINAL IFR TFC; BUT WAS KNOWN VFR TFC CLBING TO 17500 FT. WE HAD RECEIVED NO PREVIOUS CALLOUT ABOUT THIS TFC. WE ALSO NEVER SAW THE PREVIOUSLY RPTED IFR TFC. SUPPLEMENTAL INFO FROM ACN 566288: DSNDING INTO AUS CLRED TO 6000 FT; ASKED TO TURN L 20 DEGS AND EXPEDITE THROUGH 16000 FT. LEAVING 16000 FT; GOT TA AT -1700 FT; FOLLOWED QUICKLY BY 'CLB; CLB.' I HAD ACQUIRED THE ACFT AND FELT THAT IF WE WERE TO STOP DSCNT AND TRY TO CLB WE WOULD HAVE LEVELED AND REVERSED DSCNT AT APPROX THE OTHER ACFT'S ALT. I ROLLED APPROX 45 DEGS R BANK AND INCREASED DSCNT RATE. THE ACFT PASSED TO OUR L AND SLIGHTLY HIGH. IT APPEARED TO BE A BEECH JET. ATC ADVISED THAT THE ACFT WAS VFR AND NOT THE ACFT WE HAD INITIALLY VECTORED TO AVOID. AS WE PASSED; THE OTHER ACFT BEGAN A BANK TO AVOID US.