A320 FLT CREW CHALLENGE ZMA FOR HDG CHANGE DUE TO WX RADAR INDICATED TSTM ACTIVITY. INITIALLY DENIED BY ZMA; FLT CREW THREATEN TO DECLARE EMER BEFORE ZMA RELENTS AND REQUESTS ADVISORY WHEN THEY CAN PROCEED BACK ON COURSE.

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IN LEVEL FLT; ATC WAS ASSIGNING HDG DUE TO VOLUME OF TRAFFIC LEAVING FL AND A LINE OF LEVEL 3 - 4 TRWS; WITH ASSOCIATED TURB. I WAS ASSIGNED A HDG OF 350 DEGS BY ATC. MY RADAR SHOWED A LEVEL 4 TRW ON THIS HDG. I ASKED CTLR HOW LONG I WOULD BE ON THIS HDG. HE REPLIED FOR AWHILE DUE TO TFC AT FL270 OFF MY RIGHT. I ADVISED ATC THAT THIS HDG WOULD TAKE ME INTO A CELL I WAS SHOWING ON MY RADAR. CTLR DID NOT ACKNOWLEDGE MY XMISSIONS AFTER THAT. I CALLED SEVERAL TIMES REQUESTING A TURN OF 20 DEGS TO THE R TO A HDG OF 010 DEGS. ATC DID NOT ANSWER ANY OF MY CALLS. I FINALLY TOLD ATC THAT I WOULD HAVE TO DECLARE AN EMER AND THAT WE WERE TURNING TO A HDG OF 010 DEGS. ATC ANSWERED AND SAID I COULDN'T TURN DUE TO TFC OFF TO MY R AT FL270. I TOLD HIM I COULD NOT CONTINUE ON HIS ASSIGNED HDG. WE TURNED AND I ADVISED ATC OF HDG. HE ROGERED MY CALL AND ASKED ME TO PROCEED DIRECT TAYLOR WHEN CLEAR OF TRW AND WX LINE. AFTER ABOUT 20 MILES ON 010 DEGS HDG; WE CLEARED THE LINE OF WX. I ADVISED ATC WE WERE DIRECT TAYLOR AND PICKING OUR SPEED BACK. I GUESS CTLR THOUGHT THAT BY NOT ANSWERING; I WOULD STAY ON HDG AND FLY THROUGH LEVEL 4 TRW AND ASSOCIATED MODERATE TURB. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR STATED ZMA CTLR ASKED IF HE COULD CLB TO A HIGHER ALT. RPTR TOLD ZMA UNABLE DUE TO OPERATIONAL RESTRICTIONS THEY HAD ONLY ONE OPERATIONAL BLEED AIR PACK. THE RPTR AGAIN ADVISED ZMA OF A NEED FOR A HDG CHANGE. ZMA TOLD THE RPTR THAT HE WAS THE ONLY ONE WANTING A HDG CHANGE AND TO MAINTAIN PRESENT HDG. RPTR ADVISED ZMA THAT COMPANY PROVIDES ADDITIONAL WX TEXT; STATING THAT HE WAS ABOUT TO ENTER AN AREA OF TSTM ACTIVITY WHICH HE COULD VERIFY ON HIS WX RADAR; AND HE NEEDED TO TURN. RPTR STATED THAT HE GAVE ZMA ANOTHER MINUTE TO RESPOND; AND ZMA DID NOT. THE RPTR THEN XMITTED ON FREQ THAT IF NEEDED; HE WOULD DECLARE AN EMER; BUT HE WAS TURNING TO AVOID WX. IT WAS AT THAT TIME THAT ZMA RESPONDED. 'YOU DON'T REALLY NEED TO DO THAT; DO YOU? THERE'S JUST TOO MUCH PAPERWORK INVOLVED. OK; JUST ADVISE ME WHEN YOU CAN TURN BACK TO TAYLOR (VORTAC). VERIFY YOU'RE NOT DECLARING AN EMER.' THE RPTR ALLEGES HE HEARD OTHER ACFT ON THE FREQ COMPLAINING ABOUT TSTM ACTIVITY AND TURB; SO HE KNEW HE WAS NOT THE ONLY ONE EXPERIENCING PROBS. RPTR ALLEGES THAT HE INCURS THIS PROB LACK OF CTLR RESPONSE FOR WX DIVERSIONS SPECIFICALLY IN ZMA AND ZJX AIRSPACE; AND CAN NOT REMEMBER THIS 'LACK OF CONCERN IN ANY OTHER PART OF THE COUNTRY.'