I managed to tunnel out today and did some experiments with Torqeedo charging on the go. One disappointment was that I couldn't get the step-up controller to work with my 28v bike battery without blowing the 4A fuse. I also risked trying it with a 5A fuse, just in case it was on the edge, and it blew that too. The combination worked when topping up an almost fully-charged Torqeedo battery but not, it seems, when really needed. Anyone know of a cheap 12-36v current-limiting device (other than a fuse)?
Other results:
Direct connection to a 12v battery (with fuse and diode) - 25W
12v battery through step-up controller to charge at 30v - 65W
28v bike battery through step-down controller to charge at 14v - 25W. This could be increased (the current is adjustable) but even 25W generates quite a lot of heat.
So the weird-looking part of the mystery object turns out not to have been very useful, except perhaps for part-charging overnight. The fastest recharge at 65W is about the same as the standard (slow) mains charger that comes with the motor.