NMAC BTWN AN SMA SEL ON DOWNWIND LEG AND ANOTHER ACFT CLBING THROUGH HIS FLT PATH AT A CTLED ARPT.

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FXE WAS LNDG/DEPARTING RWY 26/31 SIMULTANEOUSLY. I CALLED TWR THE FIRST TIME 30 MI W IN 3000 FT HDG DIRECT FXE; AND WAS TOLD TO ENTER A R DOWNWIND FOR RWY 26 AND RPT MIDFIELD SQUAWKING 1200. THE CTLR WAS VERY BUSY; SO I DECIDED TO GIVE HIM A NEW POS RPT 10 MI W; WHICH I DID. THE CTLR AGAIN TOLD ME TO ENTER A R DOWNWIND RWY 26; BUT THIS TIME IN A RATHER HARD TONE. I TURNED ALL LIGHTS ON ON THE AIRPLANE; BECAUSE IT WAS 10-15 MINS AFTER SUNSET AND WE WERE CLOSE TO A VERY BUSY ARPT (NAV LIGHTS; BEACON; STROBES AND LNDG AND TAXI LIGHTS). I WAS PAYING CLOSE ATTN TO THE CONVERSATION ON THE RADIO; WHILE SCANNING THE OUTSIDE AREA FOR TFC. I SUDDENLY SAW AN AIRPLANE ON MY 1:30 O'CLOCK POS CLBING THROUGH MY ALT. DISTANCE WAS 1/4 MI MAX. I PUSHED THE YOKE FULL FORWARD WITH THE PWR IDLE AND WAS ABLE TO AVOID THE OTHER ACFT BY A FEW FT. THE OTHER ACFT HAD NO LIGHT ON; NOT EVEN A BEACON. HE DID NOT TAKE ANY EVASIVE ACTION; AND CONTINUED HIS CLB AS IF NOTHING HAPPENED. AFTER I ADVISED THE CTLR THAT I JUST MISSED ANOTHER ACFT BY A FEW FT; THE ONLY REPLY I GOT WAS 'CLRED TO LAND RWY 26.' I DO NOT KNOW IF THE ACFT WAS TAKING OFF RWY 31; OR WAS ON THE MISSED APCH ILS RWY 8. I DID NOT SEE THE OTHER TFC BECAUSE IT WAS NEARLY DARK; HE HAD NO LIGHTS; AND HE MIGHT HAVE BEEN COVERED BY THE DOOR POST.