A B737-300 APCHING DCA RWY 19 WAS DELAYED FOR A FLYBY. DURING A TWR DIRECTED GA THE CREW DECLARED MINIMUM FUEL AND WAS RECLRED THE APCH ONLY TO GO NEAR P56 ON SHORT FINAL.

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WE WERE FLYING TO DCA AND EXPECTING THE RIVER VISUAL TO RWY 19. AT OUR ARR TIME; TFC WAS BEING DELAYED TO ACCOMMODATE A FLYBY AT ARLINGTON. THIS DELAY USED ABOUT 1000 LBS OF OUR RESERVE. ONCE TFC WAS BEING CLRED TO DCA; A PRIORITY DEP WAS ANNOUNCED WITH SOME ACFT; OURS INCLUDED; BEING VECTORED TO TRY AND GET US ON THE GND PRIOR TO THE DEP. WE WERE CLRED FINAL AND WERE PASSING 1800 FT ON THE ARR WHEN WE WERE TOLD WE WOULD HAVE TO HOLD. WE HAD REACHED OUR BINGO AND I TOLD TWR TO SEND US TO IAD. THEY THEN DIRECTED US TO CLB AND TURN R; WE DID SO AND GAINED ABOUT 1500 FT. THEY THEN CHANGED THEIR MIND AND CLRED US TO CONTINUE THE APCH. WE TRIED BUT WERE UNABLE TO GET THE ACFT DOWN TO ALLOW A STABILIZED APCH. WE WENT AROUND AND I DECLARED MINIMUM FUEL. THEY THEN CLRED US TO PULL UP FOR A R DOWNWIND TO LAND RWY 19. AS WE WERE TURNING; WITH VARYING VECTORS FROM TWR; THE INSIDE DOWNWIND ENDED UP PLACING US CLOSER TO THE OFFSET FINAL THAN WE SHOULD HAVE BEEN. WHEN THE FO TRIED TO MAKE THE TURN FROM THE INSIDE DOWNWIND POS; TURNING ROOM WAS MINIMAL AND WE OVERSHOT IN THE AREA OF P-56. I LATER CALLED THE TWR TO ASK ABOUT OUR GND TRACK; THEY STATED OUR OVERSHOOT WAS ACTUALLY BTWN THE 'A' AND 'B' AREAS OF P-56; NO INTRUSION OCCURRED. I; AS THE CAPT; ALLOWED THE WORKLOAD OF BOTH THE FLYING FO AND MYSELF TO BECOME TOO MUCH IN A CRITICAL PHASE OF FLT. THE FO WAS RECENTLY RETURNED; WHILE HE IS AN EXCELLENT AVIATOR; I SHOULD HAVE ELECTED TO FLY THE LEG AS A REORIENTATION OR; SHOULD HAVE BEEN MUCH MORE DIRECTIVE IN THE ACTUAL FLYING OF THE MISSED AND REAPPROACH. THE ACFT WAS HANDLED IN A SAFE MANNER THROUGHOUT; THE GND TRACK JUST WASN'T WHAT WE WOULD HAVE PREFERRED. NO MENTION WAS MADE OF EITHER THE FLYBY OR THE PRIORITY DEP IN THE NOTAMS OR ATIS. I ADMITTEDLY FELT PRESSURE TO GET THE AIRPLANE ON THE GND DUE TO FUEL. FUEL CONSERVATION IS NICE; BUT IF THE OP DOESN'T IDENT THINGS LIKE FLYBYS AND PRIORITY DEPS I'M THINKING I'LL BE ADDING GAS ON MY FUTURE DCA LEGS.