NMAC BTWN AN LTT DSNDING FOR AN APCH AND AN SEL SMA PASSING THROUGH THEIR PATH. TCASII WARNING.

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CLRED FOR ILS RWY 28 GNV DANTE TRANSITION; DSNDED DOWN TO 2000 FT ON THE DANTE TRANSITION; SWITCHED TO GNV TWR AFTER LOC CAPTURED. (NO OTHER TFC RPTED IN AREA BY ZJX.) AFTER WE CHKED IN WITH GNV TWR; WE RECEIVED TA ON TCASII AT THE SAME TIME GNV TWR IS GIVING OUT TA TO THAT ACFT ABOUT OUR POS. TA INDICATES ACFT AT SAME ALT AND SAME POS AS US ON THE 5 MI SCALE ON THE TCASII DISPLAY!! AT THE SAME TIME WE ARE GOING IN AND OUT THRU CLOUDS; I NOTICED AN ACFT MOVING FROM 12 O'CLOCK TO 4 O'CLOCK RELATIVE POS TO US; SO I BANKED OUR ACFT 45 DEGS DSCNT TO THE L TO GET AWAY FROM TA ACFT; WE NEVER SAW THE ACFT. WE RECENTLY HAD TCASII INSTALLED IN OUR ACFT RIGHT WHERE WE USED TO KEEP THE CHKLIST. AS AN OLD HABIT WE SOMETIMES PUT CHKLIST WHERE TCASII IS NOW AND IT COVERS TCASII DISPLAY. THAT'S WHAT HAPPENED THIS TIME. OUR COMPANY IS TOO CHEAP TO INVEST IN RACKS OR HOLDERS FOR CHKLISTS; AND OF COURSE THEY WOULD NEVER INVEST IN INDIVIDUAL TCASII DISPLAYS; IE; VS1 DISPLAYS. ANOTHER FACTOR: RADAR COVERAGE BELOW 2000 FT AROUND GNV IS VERY LIMITED AND ZJX IS OVERWORKED IN THIS SECTOR COVERING 3-5 FREQS. OTHER ACFT INVOLVED OBVIOUSLY VIOLATED BASIC VFR CLOUD CLRNC. TYPE OF ACFT I THINK WAS A PA28 OR SIMILAR.