MLG CREW IGNORES TCASII RA AND THEN HAS AN NMAC WITH THE SMA THAT TRIGGERED IT.

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ZHN HANDED US OFF TO KONA TWR ADVISING US OF AN MLG ABOUT TO TKOF. NO OTHER TA'S FROM CTR. SWITCHED TO KONA TWR AND RPTED 12 MI OUT; MIDFIELD AND TURNING DOWNWIND FOR RWY 35. OUR ALT WAS APPROX 3500 FT. JUST AS THE TWR WAS ACKNOWLEDGING OUR INITIAL CALL-UP; THE TCASII ACTIVATED A WARNING WITH THE COMMAND TO 'CLB; CLB.' OUR INITIAL REACTION TO THE TCASII WARNING WAS THAT IT WAS RESPONDING TO THE ACFT THAT COINCIDENTALLY HAD JUST ROTATED OFF THE RWY; AND THAT IT WAS RESPONDING TO THAT ACFT. HOWEVER; ABOUT 3-4 SECONDS LATER; I OBSERVED A LIGHT SINGLE ENG ACFT (PROBABLY AN SMA) AT OUR 12 O'CLOCK POS FLYING DIRECTLY TOWARDS US AND ABOUT 1/2 MI DISTANCE AND 500 TO 600 FT BELOW OUR ALT. WE WERE IN ABOUT A 1000 FPM DSCNT AT THE TIME. I CALLED THE TFC TO ATTN OF FO WHO WAS FLYING THE ACFT; ALONG WITH THE COMMAND 'HOLD YOUR ALT.' HE RESPONDED WITH BACK PRESSURE ON THE CTLS AND I ALSO GRABBED THE CTLS AND PUT ADDITIONAL BACK PRESSURE ON THE CTLS AND TRANSITIONED THE ACFT INTO A 500 TO 1000 FPM CLB. THIS TRANSITION WAS AT A RATE THAT CAUSED ONE OF THE FLT ATTENDANTS TO LATER COMPLAIN THAT IT CAUSED HER KNEES TO BUCKLE AND SHE ALMOST DROPPED TO 1 KNEE. THE LIGHT ACFT APPEARED TO BE IN A SHALLOW CLB AND DID NOT ALTER OR CHANGE ITS FLT PATH OR ACFT ATTITUDE IN ANYWAY AND GAVE EVERY INDICATION THAT HE DID NOT SEE US. ABOUT 5 TO 6 SECONDS LATER; WE PASSED DIRECTLY OVER THE ACFT WITH ABOUT 500 FT OF VERT SEPARATION. I THEN ASKED THE TWR IF HE HAD ANY TFC IN THE AREA AND HE SAID YES; A LIGHT ACFT DEPARTING THE AREA. THE OBVIOUS QUESTION IS: WITH A RWY 35 AS THE ACTIVE RWY; HOW OR WHY IS A LIGHT ACFT CLBING OUT OF AN ARPT AREA DIRECTLY INTO THE APCH CORRIDOR USED BY ALL THE COMMERCIAL JET TFC INTO THIS ARPT WHEN RWY 35 IS IN USE?? THIS ACFT WAS IN CONTACT WITH KONA TWR IN THE TIME FRAME LEADING UP TO THIS POINT. KONA TWR WAS ALSO WELL AWARE OF OUR APCH INTO THE AREA. I DO NOT KNOW IF TWR ADVISED THE LIGHT ACFT OF OUR APCH; HOWEVER; WE WERE NOT ADVISED OF THE TFC BY CTR (WHO I'M SURE DID NOT KNOW OF THE ACFT) OR BY THE TWR; WHO SURELY DID KNOW. I THINK THAT THE TWR SHOULD BE DISCOURAGING LIGHT ACFT DEPARTING OR TRANSITING THE AREA AWAY FROM THE APCH CORRIDOR AND DOWNWIND PATTERN OF INCOMING COMMERCIAL JET TFC!