I'm still interested that the original post seems to imply that this noise is NOT present on a clean amp.
I know that gain increases the effect of buzzes and hums, but I would still expect to be able hear them through a clean amp. I would not expect to be able to say the following if I had an electrics issue of some sort in the guitar itself:
"Now, the pickup sounds just fine clean, both full hum and coil tapped. Just rolling the tone knob off and on between tapped and hb is like toggling between pickup combos with a single coil and humbucked guitar. Sounds great."
The other thing I'm wondering is how the pickup has been hooked up. Now, I haven't had a humbucker with 4 wires for multiple configurations for many years, so I'm no expert, but I do remember that you get strange effects if you don't get the hook-ups quite right (especially if it's a covered pickup? can't remember). Anyway, I read a lot on here that the BKP colour scheme for wiring is different than some big manufacturers use - is it possible the tech is not aware of this? (I doubt it, if he's already used BKPs)
I don't know though, I'm just chucking out the questions I'd have in my head if the problem was mine.