CL30 FLT CREW FAILED TO CROSS LAX AT FL430 AS PER ATC AND CONFLICTED WITH OPPOSITE DIRECTION TFC AT FL410.

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DEPARTED SEE AND CLBING TO CRUISE ALT. ATC CLRED US DIRECT LAX AT ABOUT 40-50 MI S OF LAX; AND CLRED US TO CROSS LAX AT FL430. AS WE NEARED LAX; WE REALIZED WE COULD NOT MAKE THE XING RESTR; AND ADVISED ATC AT 10-15 MI S OF LAX. ATC ASSIGNED US A HDG OFF COURSE; THEN REVISED TO REMAIN ON COURSE; WHILE POINTING OUT OPPOSING TFC AT FL410. TFC WAS SIGHTED AND ACKNOWLEDGED; AND PASSED OFF OUR L AT FL410 AND ABOUT 1-2 MI. WE WERE AT FL425 WHILE PASSING AND STILL CLBING. ATC REQUESTED A PHONE CALL AFTER LNDG AND RPTED THAT LOSS OF SEPARATION HAD OCCURRED AND THE INCIDENT WOULD BE REVIEWED AND INVESTIGATED FOR POSSIBLE ENFORCEMENT ACTION. THIS SITUATION COULD HAVE BEEN PREVENTED BY NOT ACCEPTING A CLBING XING RESTR AT SUCH HIGH ALTS; WHERE CLB PERFORMANCE BEGINS TO DECREASE. I SHOULD HAVE RECOGNIZED SOONER THAT WE MIGHT NOT MAKE IT AND TAKEN POSITIVE ACTION TO CHANGE THE SITUATION. IT WOULD HAVE BEEN HELPFUL IF ATC HAD ADVISED INITIALLY OF THE XING TFC; AND ALSO IF ATC HAD QUERIED US ON OUR ABILITY TO MEET THE RESTR AS WE GOT CLOSER TO THE VOR AND THE OPPOSING TFC. THERE WERE NOT ANY TCAS WARNINGS GENERATED BY THIS EVENT EITHER.