ACR LGT PENETRATED RESTRICTED AREA WHILE DEV FOR TSTM. ARTCC RADAR CTLR ADVISED THE FLC AFTER THE FACT.

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FLT DEPARTED JAX ENRTE TO BNA AT XA56Z. AFTER TKOF FROM RWY 13 AT JAX; A L TURNOUT WAS ISSUED BY ATC. NUMEROUS TSTMS WERE IN THE AREA TO THE N AND W OF JAX ARPT. DEVS WERE AUTHORIZED BY ALL CTLRS UP TO TIME OF INCIDENT. THE ORIGINAL ENRTE CLRNC WAS DIRECT ALMA; GA (AMG). DUE TO THE WX; WE WERE FORCED TO DEVIATE N AND E OF COURSE. AT APPROX XB25Z WE WERE HANDED OFF TO JAX CENTER (135.45). ON INITIAL CONTACT WE WERE ISSUED A 360 DEG HDG. WE RESPONDED THAT WE WERE UNABLE TO COMPLY AND WE NEEDED TO CONTINUE TO DEVIATE TO THE N AND E FOR WX AVOIDANCE. AFTER A FEW MINS PASSED; JAX CENTER ASKED US TO SQUAWK 7700. NOT TELLING US WHY HE WANTED US TO SQUAWK 7700; WE THEN ASKED WHY AND HIS REPLY WAS THAT WE WERE IN A RESTR AREA - THAT IS HOT - AND HAS LIVE FIRE AT THIS TIME. WE THEN ASKED WHY WE WERE NOT NOTIFIED BEFORE THIS POINT. THE CTLR RESPONDED THAT HE JUST INFORMED US. AT XB30Z WE WERE ABLE TO ACCEPT A 330 DEG HDG; THROUGH A BREAK IN THE LINE OF TSTMS. OUR POS AT THIS POINT WAS APPROX 35 NM NE OF BRUNSWICK VOR. WE WERE THEN ISSUED A NEW SQUAWK AND CHANGED TO JAX CENTER (126.12). FLT CONTINUED TO BNA WITHOUT FURTHER INCIDENT. I CONTACTED DISPATCH UPON ARR AT BNA TO DEBRIEF THE FLT. PREVENT THIS OCCURRENCE: JAX CENTER CTLR HAD TOO MANY ACFT IN HIS SECTOR. UNABLE TO DEVOTE MORE TIME TO US. SINCE WE WERE BOXED IN; WX N AND W OF US; ONLY WAY TO GO WAS E THEN N. SHOULD HAVE ADVISED US SOONER THAT THE ONLY WAY UP THE E COAST OF FLORIDA WAS TO GO THROUGH RESTR AREA TO AVOID WX; WE THEN COULD HAVE OPTED TO GO S AND W THEN N AFTER DEP FROM JAX TO AVOID WX.