It turns out that in the LaCie EBD the 2nd drive has data right at the start in sector 0 (I did claim otherwise before, but I was "speaking from my memory" and it let me down that time). I also dug up a "success story" on another forum about the Big Disk coughing up the user's data after the last partition on the first drive and the (only) partition on the second drive were simply concatenated. That's JBOD. I noticed that the RAID controller

likes to wipe out sector 0 on the drives it touches. Sure enough my corrupted drive 2 had 512 bytes of 0's in sector 0 and random-looking data starting in sector 1 and going on forever. The "good" drive 2 had random-looking data starting from sector 0 and going forever. The data appears to be a part of some file that just happened to be stored at the stitch between the 2 drives. The data in these same few sectors at the start of drive 2 doesn't match at all between the bad and the good drives and the partitions are set up the same way so if it was directory-structure-related I would have found some correlation at least. Everything else the RAID controller touches seemed to be in the sections of the drives that are not readily user-writable and the sectors could be copied from the good drives with high probability of success since the surrounding sectors matched exactly on both drives. Sure enough, when I threw the fixed 2 drives into the enclosure they worked as if nothing had happened. Sector 0 on drive 2 is lost, however, because it apparently is a part of
my file, so I couldn't just copy it from the good drive's. To verify that I want to find which file it is and whether or not it's visibly affected. It's possible that I have a backup of that particular file somewhere, too.
In any case, despite my modest abilities I am much better off than last week and it didn't take that much effort or resources, plus I now know what the RAID controller messes with and got exposure to some of the tools that help examine, tinker with, backup or fix drives.
Thanks for your help