CLOSE PROX ACR MLG ON SID OUT OF LAX AND UNK VFR ACFT INBOUND.

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INCIDENT OCCURRED WHILE CLBING OUT OF LAX ON THE GORMAN 1 SID; WITH CLRNC TO CLB TO FL230. POS AT THE START OF THE CONFLICT WAS OUTBOUND ON THE LAX 328 RADIAL; 25 DME. PASSING THROUGH ABOUT 14100 FT; TCASII ANNOUNCED 'TFC; TFC.' TARGET APPEARED TO BE AT ABOUT 15500 FT IN MY 1 O'CLOCK POS; CLOSING. A COUPLE OF HUNDRED FT LATER IN THE CLB ATC INSTRUCTED ME TO LEVEL AT 15000 FT FOR TFC; AND SIMULTANEOUSLY TCASII ORDERED ME TO 'CLB; XING CLB' DEMANDING A RATE IN EXCESS OF 2500 FPM. SINCE I HAD TAKEN OFF JUST 2000 POUNDS SHY OF MY MAX TKOF WT; I WAS STILL FAR TOO HVY TO MAKE SUCH A HIGH PERFORMANCE CLB; SO I CHOSE TO IGNORE TCASII AND COMPLY WITH ATC'S INSTRUCTIONS TO LEVEL AT 15000 FT. I ALSO STARTED A 15 DEG BANK TURN TO THE L TO AVOID THE TARGET'S FLT PATH. AS I WAS PASSING THROUGH 14800 FT I HEARD THE OTHER ACFT SAY THAT HE WAS TURNING R. SINCE THIS WOULD AGAIN PUT HIM ON AN INTERCEPT WITH ME; I REVERSED MY TURN; ROLLING OUT IN A 15 DEG BANK R TURN. WHILE I WAS DOING THIS; TCASII CHANGED ITS MIND AND ISSUED A 'DSND; DSND' COMMAND. SINCE THIS SEEMED REASONABLE; I INITIATED A DSCNT OF ABOUT 1000 FPM; EVENTUALLY GETTING BACK DOWN TO ABOUT 14200 FT. AT THIS POINT THE TARGET WAS BEHIND ME AND THE CONFLICT RESOLVED. I TURNED BACK TO THE DEP AIRWAY AND STARTED TO CLB BACK TO 15000 FT. BEFORE I REACHED 15000 FT; ATC RECLRED ME TO FL230 AND WE CONTINUED ON OUR JOLLY WAY. DURING THE ENTIRE SEQUENCE MY HIGHEST ALT WAS 14900 FT; LOWEST 14100 FT. ACCORDING TO TCASII THE LEAST VERT SEPARATION BTWN MY ACFT AND THE TARGET WAS 700 FT AND THE LEAST HORIZ SEPARATION WAS ABOUT 1/4 MI.