B737 FLT CREW HAS TO TAKE EVASIVE MANEUVERS DURING TKOF TO AVOID ANOTHER ACFT ON A MISSED APCH.

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WE TOOK OFF ON RWY 25R AT LAS; AND CLBING THROUGH 200-300 FT AGL; TWR SAID THAT THERE WAS AN ACFT ON A GAR ON ONE OF THE 19 RWYS; AND REQUESTED THAT WE 'STAY LOW THROUGH THE INTXN.' LOOKING OVER TO THE R; WE COULD SEE A B737-200 FLYING LEVEL OVER RWY 19; APPARENTLY CO-ALT WITH US AND ON A COLLISION COURSE. NOT KNOWING IF THE FO UNDERSTOOD WHAT TWR WAS REQUESTING; I DIRECTED HIM TO LEVEL OFF; WHICH HE DID AT ABOUT 400 FT AGL; REDUCING THRUST TO MAINTAIN ABOUT 170 KTS. TWR INFORMED US THAT THE OTHER ACFT WAS GOING TO CLB; WHICH IT DID AFTER A SHORT MOMENT. THE TCAS SYMBOLOGY SHOWED THE OTHER ACFT 600 FT ABOVE US AS IT PASSED DIRECTLY OVERHEAD. THE SITUATION RAISES A NUMBER OF QUESTIONS. FOR ONE THING; WHY WASN'T THE OTHER ACFT CLBING AS PART OF ITS GAR? THE FO AND I TALKED ABOUT WHAT WE COULD HAVE DONE HAD THE OTHER ACFT NOT CLBED; AND IT DIDN'T SEEM LIKE WE HAD ANY FOOLPROOF OPTIONS. THE FO SAID HE WAS PLANNING ON TURNING L IF THE OTHER ACFT HADN'T CLBED. MY BELIEF WAS THAT A R-HAND TURN WOULD HAVE PROVIDED SEPARATION FROM THE OTHER ACFT; WHEREAS A L TURN WOULD NOT HAVE. THERE WERE TOO MANY WAYS FOR THIS TO GO WRONG -- CONGESTED RADIOS; CONFUSION OVER TWR'S INSTRUCTIONS; INDECISION AND OTHERS. THE FACT THAT THINGS WORKED OUT AS TWR DIRECTED WAS LARGELY DUE TO LUCK; AND I WOULDN'T WANT TO DEPEND ON THAT NEXT TIME. SUPPLEMENTAL INFO FROM ACN 688152: I WAS ON THE VERGE OF EXECUTING A STEEP TURN AWAY FROM THE ACFT WHEN THE OTHER ACFT SUDDENLY PITCHED UP AGGRESSIVELY AND STARTED TO CLB.