HVY WX ENCOUNTER CAUSES LOST COM.

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OUR ACFT WAS IFR; SBOUND ON V-437 AT 8000 FT. A LINE OF HVY WX WAS PARALLEL TO THE AIRWAY ON THE W SIDE. WE WERE FLYING THE L SIDE. (E SIDE) OF THE AIRWAY TO AVOID TURB. ZJX INSTRUCTED US TO FLY W AND REJOIN THE CTRLINE TO AVOID A RESTR AREA. (AN APPARENT REF TO W-159 A OR 158 A. I DESCRIBED OUR REASON FOR THE CTRLINE OFFSET AND THE HVY WX PARALLEL TO OUR TRACK ON THE W SIDE. THE CTLR CALLED AGAIN WITH VECTORS WHICH WOULD CARRY US INTO THE WX AND SAID SHE HAD BEEN TOLD BY SOME OTHER PLT THERE WAS NO WX IN THE AREA. IN AN EFFORT TO MAKE HER HAPPY; I TURNED TO MOVE CLOSER TO THE CTRLINE. APPROX 1000 FT FROM THE CUMULUS BUILD UP WE FLEW INTO HVY TURB. WE HAD COMPLETE LOGS OF DIRECTIONAL CTL AND WERE UNABLE TO MAINTAIN OUR ASSIGNED ALT CLOSER THAN 1000 FT. THE TURB CONTINUED FOR 5-10 MINS. DURING THAT TIME WE LOST ALL RADIO NAV AND COM EQUIP DUE TO DISCONNECT OF CIRCUIT BREAKERS AND INTERCOM CONNECTIONS CAUSED BY HVY TURB. AFTER ABOUT 12 MINS; I WAS ABLE TO RESTORE PARTIAL RADIO NAV CAPABILITY. WE SQUAWKED; 7700 FOR 1 MIN; SWITCHED TO 7600 AND CONTINUED FOR OUR DEST ARPT (TIX). WE LANDED AT THE CTLED FIELD WITH NO INCIDENT. ATC REFUSAL TO ALLOW A SLIGHT DEV FOR WX AVOIDANCE; AND INSISTENCE THAT WE REJOIN THE CTRLINE COULD HAVE TRIGGERED A FATAL ACCIDENT. THE CONFIGN OF THE ADIZ; WARNING AREA PERIMETERS AND V-437 AT THIS POINT -- PLUS THE PROX OF A LINE OF HVY WX SHOULD HAVE DOCUMENTED OUR NEED TO ENCROACH ON THE ADIZ AND EXPLAINED OUR PRESENCE IN THE WARNING AREA. AFTER THE INCIDENT AND PARTIAL RECOVERY OF NAV CAPABILITY; WE FLEW STANDARD LOST COM PROCS; WENT TO VFR CONDITIONS AT THE FIRST OPPORTUNITY; AND LANDED AT THE FIRST VFR ARPT (TIX).