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Ok there are actually numerous issues with these fuel pumps failing .Wen the vehicle was running was the check engine light on or any dash lights blinking?
Ok was there actually an issue with the rear brake lights?As for the code for the gas cap are you sure it was not an eval code fuel the fuel system?
Ok if this was the fuel pump it would have died completely and not restarted this can even be an issue with the vehicles security system not reading the chip in the key and that will cut out the fuel pump and cause the vehicle to stall thats what it sounds like to me. Another issue is it can just be a dirty throttle body or faulty mass air flow sensor .How much do you trust the shop its in now.
Ok tell him to check the security system.its calls the immobilizer system there is a chip in the key and the ignition switch reads that chip and allows the fuel pump to operate and the vehicle to start .If there si a damaged key ,damaged ignition switch or transponder ring around the ignition switch it can cause this issue .When this fails the fuel pump will still come on its just cuts fuel to the fuel injectors.This sounds like more of the issue to me not the fuel pump .He needs to start here and check these items it can be as simple as disconnecting the battery for a half hour then using a different key if you have 2 to fix this issue.I would start here next.Hoped this helped.Please rate excellent service so I can get credit for my help thanks.Bonuses greatly appreciated.Im here if you need anything else .