RADAR CTLR ISSUES A 'GOOD RATE' CLB TO FL370 FOR B757. FLC FAILED TO ADVISE CTLR THAT THEY NOW HAD TO REPROGRAM THE FMC FOR FL370. IT THEN GOES INTO A 'RESYNC' MODE AND CLB RATE FALLS OFF. XING CRUISE TFC AT FL350 PASSES ABOVE BY 300 FT AND WITHIN 2 PT 5 MI. BOTH ACFT RECEIVED TCASII RA'S.

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ON CLB OUT OF DTW; CTLR ASKED IF WE COULD GIVE GOOD RATE OF CLIMB TO 370 FROM 310. WE ANSWERED YES; AND HE THEN CLRED US TO FL330. AT THE FL330 LEVELOFF; WE WERE GIVEN GOOD RATE TO 370 AND A TURN TO THE S FOR THE CLB; AFTER SOME HESITATION BY THE CTLR. AFTER LEVELING AT FL330 AND THE HESITATION ALONG WITH THE CLRNC THAT FOLLOWED TO FL370; THE ACFT LEVELED AT FL330 AND WE HAD TO REPROGRAM THE FL370 CLB. WE THEN BEGAN A CLB AND BROUGHT OUR SPD BACK TO INCREASE THE RATE WHILE LOOKING FOR TFC POINTED OUT TO US. THE CLB RATE PAUSED FOR A FEW MOMENTS AND I NOTICED OUR FMC'S GAVE A MESSAGE THAT THEY WERE RESYNCING. DURING THAT TIME THE RATE FELL OFF. WE THEN INTERVENED TO INCREASE THE RATE AND OUT OF FL347 WE GOT A TCASII ALERT (RED) ADVISING US NOT TO CLB. WE COULD SEE THE TARGET ACFT AND TURNED FURTHER S TO AVOID HIM; AND DISCONNECTED THE AUTOPLT AND FOLLOWED THE TCASII COMMAND. I BELIEVE THE HESITATION AND SUBSEQUENT REPROGRAM CREATED A SIT THAT MADE IT VERY DIFFICULT TO PROVIDE THE CLB NEEDED. THIS WAS COMPLICATED BY THE MOMENTARY PAUSE CAUSED BY THE RESYNC. SUPPLEMENTAL INFO FROM ACN 364934: RECEIVED RA ON TCASII WITH ACFT 1 O'CLOCK POS (SEBOUND) APPARENTLY CLBING TO FL370; STARTED CLB LEAVING FL350 (ASSIGNED CRUISE). ACFT PASSED IN FRONT ABOUT FL350 AND 3 NM. ATC ADVISED OF RA AND CLB; WAS RECLRED BACK TO FL350. THIS IS NOT CONSIDERED A NEAR MISS; ONLY LOSS OF SEPARATION.