OK. I have figured it out on my own. Posted herewith for the benefit of anyone who has the BAR-engine 4.2 FSI Touareg and also needs to replace this sensor.
You will need to remove the right front wheel and remove the fasteners from the back half of the inner fender lining. You can fold the liner forward if you don't want to remove the whole thing. Don't forget the Torx 30 down at the bottom corner facing the ground.
There's a gap near the control arms where you will be able to see into the engine compartment. You will need to stick your hands in there to remove the connector. If you happen to be a ten year old girl this will make the job a lot easier, but if you've got manly man hands then prepare for a bit of frustration and bruised wrists.
Inside this gap, you will find a very stylish pouch made from silver heat-resistant fabric, complete with two snaps, next to the engine. Pop these snaps open and remove the pouch, and this will reveal the elusive connector end of the crankshaft position sensor. Congratulations, Indiana Jones, your search for the Holy Grail has been rewarded.
The connector is mounted on a metal post. You can pry it outward off this post to give yourself a little more slack to make it easier to disconnect. But, being a VW connector, you will pry and pull and wiggle on it for about an hour, before it suddenly and magically disengages. Using as many four-letter words as you have in your vocabulary (and a small flat screwdriver) seems to help.
I tied a long string to the connector, and then pulled it out from the bottom of the vehicle. I then tied the string to the new connector and pulled it up the other way, but it jammed on what I assume are the transmission cooler lines. You'll see a 10mm bolt holding them in, down near the crank position sensor location; remove the bolt and the cooler lines will loosen up just enough for you to squeeze the connector through.
Reassemble your Touareg, fire up the engine, and bask in the warm glow of satisfaction in having saved $500 because you didn't take it to the dealer.