ZDC UNABLE TO ACCOMMODATE B737 FLC HDG CHANGE DUE TO WX ADVISING THEM TO SQUAWK 7700 AND 'DO WHAT YOU HAVE TO' AS FLC INITIATES EVASIVE MANEUVER.

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UPON CLBOUT FROM DCA AND INITIAL TURN FROM THE DCA 328 DEG RADIAL TO AN EASTERLY HEADING; PROCEEDING IN THE DIRECTION OF PALEO INTERSECTION; ATC WAS UNABLE TO ACCOMMODATE A TURN FOR HAZARDOUS WX. WE ADVISED WASHINGTON CENTER OF THE NECESSITY TO TURN NW; AND THEY DECLINED OUR REQUEST TELLING US TO; 'SQUAWK 7700 AND DO WHAT YOU HAVE TO.' WE SET THE TRANSPONDER TO 7700 AND TOOK APPROPRIATE EVASIVE WX AVOIDANCE TURNING TO A NORTHWESTERLY HEADING. THE CTLR ON WASHINGTON CENTER FREQ 124.55 STATED; 'HE WASN'T PICKING UP ANYTHING GREATER THAN LEVEL 1 ACTIVITY; AND THAT OTHER ACFT TRAVERSED THE AREA; SO WHY COULDN'T WE?' FLT CONDITIONS WERE RAPIDLY DETERIORATING; WITH CONTINUOUS MODERATE TURB AND LIGHTNING IN THE DISTANCE. TIME WAS CRITICAL SINCE ONBOARD RADAR DEPICTED CELL ACTIVITY AHEAD; AND ANY FURTHER DELAY WOULD HAVE COMPROMISED SAFETY. WE WERE APPARENTLY DENIED A TURN DUE TO OVERLAPPING SECTOR AIRSPACE RESPONSIBILITY. WE DIDN'T DECLARE AN EMER; BUT COMPLIED WITH SETTING THE TRANSPONDER TO 7700 FOR TURN INITIATION.