I would NOT be so fast to send it back.
Is this on your new Special 4H, or (as your clip name suggests) a Classic?
The Neodymium pickups on the Specials have a stronger magnetic pull than Alnico or Ceramic pups, and therefor may need to have a bit more distance from the strings to prevent them from pulling the strings out of their natural mechanical path of oscillation. The magnetic pull of the pups can change the way the string vibrates, causing overtones, chorusing, pitch warbles, etc. Since you say the sound is occuring when you are plucking in the pup's general area, perhaps you are causing the strings to swing closer to the pup by plucking there (does sound louder in your clip during those sections). Since you also mentioned that you have lowered your strings, this seems likely that you may just need to lower the pickup, and get to rockin'.