Well, you know the rest. Turned out the issue with the car was a carabiner lodged against the battery clamp, shorting the whole mess. (And if you wonder why there was a carabiner there, it's a sensible question -- the battery is normally held in by a single band of thin metal, so we had to rig up a support structure to avoid road damage while in motion. This turned out to be bungees, thus the carabiner, thus the problem.)

Thanks anyway, everyone -- at least we've finally got the car home and working. Now we just wait for the garage to finish fixing the van, and then we'll have two working vehicles again.