AFTER DEPARTING DFW; B737 CREW OBSERVED A TSTM ON THEIR COURSE AND WERE DENIED A REQUEST FOR A 10-20 DEG DEV TO AVOID THE WX.

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FLT DEPARTED FOR LIT; CLRED DALL 6.LIT TRANSITION. REGIONAL DEP HANDED US OFF TO ZFW. RADAR TURNED ON AND PAINTING WX WITH RED CENTERS ON RETURNS. CTR GAVE US A VECTOR TO INTERCEPT THE 064 DEG RADIAL OFF OF TTT VOR. WE COULD SEE THAT THE DEP RADIAL WOULD TAKE US THROUGH THE WX. WE ASKED IF WE COULD MAINTAIN OUR PRESENT HDG APPROX 030-040 DEGS TO GO AROUND WX. REQUEST WAS DENIED. WE EXPLAINED THAT THERE WAS WX TO OUR R AND ON THE DEP RADIAL AND WE NEEDED TO DEVIATE. REQUEST WAS AGAIN DENIED. WE WERE TOLD THERE WERE ARRS AHEAD AND AFT OUR L AND IT WAS NOT THIS CTLR'S AIRSPACE. WE ASKED; 'IF WE EXPEDITE OUR CLB CAN WE GET ABOVE THE ARRS AND DEVIATE?' WE WERE TOLD 'NO; TURN R AND JOIN THE RADIAL.' WE ONCE AGAIN TOLD THE CTLR WE COULD NOT TURN R BECAUSE OF THE TSTM. WE TOLD THE CTLR WE COULD MAKE A 180 DEG TURN AND GO BACK TOWARD DALLAS AND DO WHATEVER HE WANTED TO GET US AROUND THE WX. WE WERE TOLD TO TURN R 180 DEGS. WE ONCE AGAIN TOLD THE CTLR WE COULD NOT ACCEPT A R TURN. AT THIS POINT; I TOLD THE CTLR WE WERE DEVIATING L OF COURSE. UPON LNDG AT LIT; THE CAPT CALLED COMPANY ATC AND RELAYED OUR EXPERIENCE. HE CALLED ZFW AND THEY REVIEWED THE TAPES. THE CTLR WAS PULLED FROM HIS POS. LOOKING BACK; THE ONLY THING I WOULD HAVE DONE DIFFERENTLY WOULD BE TO DECLARE AN EMER AND EXERCISE EMER AUTH. SUPPLEMENTAL INFO FROM ACN 536039: TIME AND POS RELATIVE TO THE WX FORCED US TO MAKE A DECISION TO HOLD A 050 DEG HDG FOR APPROX 5 MI. IF WE HAD BIGGER ENGS AND MORE TIME; WE COULD HAVE TOPPED THE WX. WE WERE FORCED TO SPLIT 2 SMALL CELLS THAT WERE GROWING AND SHOWING RED CENTERS; WHILE AVOIDING THE WX OVER THE OUTBOUND RADIAL.