SMA ACFT ON IFR TRAINING FLT EXPERIENCED NMAC ON GPS RWY 13 APCH TO LEE; ALLEGING MCO TFC CONGESTION AND LATE FREQ CHANGE AS CONTRIBUTORY.

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OUR ACFT; AN AEROSTAR TWIN WAS ON AN INST INSTRUCTION FLT WITH MYSELF (INSTRUCTOR) AND ACFT OWNER (COMMERCIAL PLT) ABOARD. WE WERE EXECUTING THE SECOND OF 2 GPS RWY 13 APCHS TO LEE ARPT. FREQ CONGESTION WITH ORLANDO APCH ON 121.1 WAS EXTREME. ORLANDO CTLR EITHER FORGOT OR WAS TOO BUSY TO TURN US OVER TO LEESBURG TWR FOR OUR REQUESTED LOW APCH TO BE FOLLOWED BY RETURN TO ORLANDO APCH. FINALLY WE REALIZED WE SHOULD BE TALKING TO LEE TWR; SO AT ABOUT 2 MI FROM THE RWY; WE SWITCHED TO THE TWR ON OUR OWN; BEFORE WE COULD CALL LEE TWR WE SAW; SLIGHTLY TO OUR R AND SLIGHTLY ABOVE US; A C172. WE BOTH TOOK EVASIVE ACTION AT THE SAME TIME; HE GOING UP AND RIGHT AND US GOING DOWN AND L. THE TWR SCOLDED US FOR NOT CONTACTING THEM SOONER. WE THEN BROKE OFF THE APCH AT 800 FT MSL AND TOLD THE TWR WE WERE DEPARTING TO THE E. THE TWR ADVISED US TO CONTACT ORLANDO APCH; WHICH WE DID; AND REQUESTED AN ILS RWY 9 APCH TO SFB TO A FULL STOP. ORLANDO APCH WAS STILL EXTREMELY BUSY ON 121.1 AND DID NOT TURN US OVER TO SFB TWR UNTIL INSIDE THE OM.