Because of where BLT is located within WDW property I don't think anything is "close".
We still go out to the Publix on Vineland (used to go to Winn Dixie but learned better). I think there is a Publix towards Windermere if you wanted to venture out that way

and another out past AK (off 192 ?) but I would expect it to be just as far if not farther.