Are you using the screws and springs that came with guitar, or the ones that BKP supplied? If the latter, I would try using the ones that were in it, if they are clearly different lengths.
It sounds a bit like you are lacking room inside somewhere and something is bottoming out, pushing the pickup upwards. Check also that those washers are not fouling the pickup. If they are rubbing somehow try cutting a bit of them with some snips.
I say this as someone who's never pulled apart a Tele, but from your description those are the first things I'd investigate.