Well, it's a Japanese alder tele with maple+rosewood neck. I has 'woody' sound when unplugged. It's difficult to say about the character of plugged sound, because the guitar has cheap and poor-sounding pickups in it.
I need the sound that lies somewhere between P90, tele single and PAF humbucker. I've managed to get the closest sound from DiMarzios Virtual Hot T and Virtual Solo. But I don't like the stacks. In fact I want less trebles, more bass and low-mids and increased output (in comparison to traditional tele pickup).
Hmm... Kinda reminds me what I get from the additional position on the 4-way switch. It's "both pups, series", which is part of the humbucker spec (2 coils series), but it's not quite a 'bucker tone, because here we have coils that are differently voiced and more distant from each other.
With the BGF50s, the resulting tone is indeed darker (less trebles and some comb filter effect in the mids) and fattier (more lows, more overall output level), resulting in a tone that really "lies somewhere between P90, tele single and PAF humbucker".
So it somehow seems to fit your bill - and the good news is that you get this tone as a bonus to the standard vintage Tele tones. OTHO, you only have one fatter tone - not a fatter bridge + a distinct fatter neck, so I can't tell if such a solution would work for you - nor even if you'd like that tone.
I don't have any even half-decent recording gear yet so I didn't post any clip, but if you're not in a hurry I could eventually try to borrow some stuff from a friend to record a quick demo of the different tones I can get with this setup. Now don't expect anything until next week... So let me know.
Can't comment on the Piledriver - at least not from direct experience. General consensus is that it has a more modern edge, and Tim kind of implied I'd rather avoid it if I was after that OneTrueVintageTeleTwang(tm) tone - but since going for a PAF was one of your options I guess you don't care that much, so well, why not ? At least there are a couple PD clips around. And it might work better with downtuned guitar, don't know...
Also, since you seem ok to do some rerouting for a PAF, you could as well think about a MQ (PAF-sized P90-like) ?
My 2 cents...