PLT ON FA20 ON ILS INTO LRP WAS UNABLE TO OBTAIN A STRAIGHT-IN ON RWY 8 AND ENTERED PATTERN FOR LNDG ON RWY 26. WHILE MANEUVERING; THE CREW RECEIVED A TCASII RA.

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WE HAD BEEN CLRED FOR THE ILS RWY 8 CIRCLE-TO-LAND ON RWY 26. UPON CONTACTING THE TWR AT THE OM; THE FO REQUESTED A STRAIGHT-IN LNDG TO RWY 8 AS PER OUR PRE-APCH DISCUSSION. THE TWR CTLR INSTRUCTED THE FO TO RPT AT 3 MI FOR A POSSIBLE STRAIGHT-IN LNDG CLRNC. AT 3 MI OUT; WITH THE RWY IN SIGHT; THE FO ATTEMPTED TO MAKE THE 3 MI RPT; BUT WAS UNABLE TO GET THROUGH DUE TO FREQ CONGESTION. AT THAT MOMENT; I NOTICED TFC 400 FT BELOW; AND 2 MI AHEAD ON THE TCASII. I WAS UNABLE TO MAKE VISUAL CONTACT WITH THE TFC IN SPITE OF CLR VISUAL CONTACT WITH THE RWY. I DECIDED TO BREAK OFF THE APCH WITH A R TURN TO ENTER A L DOWNWIND LEG FOR RWY 26. AS I BEGAN THE R TURN; THE TCASII COMMANDED A 'CLB' RA. I IMMEDIATELY APPLIED PWR AND INITIATED A STEEP CLB. NEITHER OF US VISUALLY SAW THE ACFT; HOWEVER; THE FO OBSERVED ON TCASII THAT WE PASSED OVER THE TFC BY 100 FT. I HAD PREVIOUSLY BRIEFED THE FO ON THE DSCNT TO MAKE OUR REQUEST FOR A 'STRAIGHT-IN APCH TO RWY 8; TFC PERMITTING' WITH APCH CTL SO THINGS COULD BE COORD IN ADVANCE. HE INSTEAD STATED HE WOULD RATHER DO SO WITH THE TWR CTLR; TO WHICH I AGREED TO DO. THIS CONFLICT BEGAN WHEN OUR 'STRAIGHT-IN' REQUEST WAS MADE AT THE LAST MIN TO A VERY BUSY TWR CTLR WITH LITTLE TIME TO SORT THINGS OUT. OUR REQUEST SHOULD HAVE BEEN IMMEDIATELY DENIED. I SHOULD HAVE INSISTED ON COORDINATING THE REQUEST; 'STRAIGHT-IN; TFC PERMITTING' WITH APCH CTL.