I would agree that it's the Greasebucket, I have it on my American Special. You'll have to dial in the tone with your amp eq for increased bass. I back off the treble, lower the gain, increase bass and mids and sounds fine. From there, the tone on the bass just smooths out any excess brightness.
Stanley, just an FYI, but your AmSpec P may likely also be using the 61276/7 set of pups as well as the later Hwy Ones ... I would likely suspect them to possibly be an issue of more signifiance than the GreaseBucket tone circuitry ...
JMHO as always ...