I emailed BJ's about it. Here is their response:
Thank you for contacting BJ'S Member Care with your inquiry.
Depending on the trade area, BJ'S buys unbranded gasoline from many different suppliers, be they national brand refiners or smaller independent refiner/suppliers. Though BJ'S purchases from many sources to maintain our low prices, please be assured that BJ'S gas maintains the strict quality guideline established by the Federal Government's Environmental Protection Agency. At all times, though, BJ'S gas maintains the strict quality guidelines established by the federal government's Environmental Protection Agency (EPA). We sell 87 Octane Regular Unleaded and 93 Octane Premium Unleaded. BJ'S gasoline contains all detergent additives required by EPA guidelines.
Still have no idea who supplies it. I may try it anyway, but it's hard to switch from name brand fuel. My truck does very well on the new Shell gas with nitrogen.