PROBLEM SOLVED!!!
Thanks to debugging versions of the plugin, Pavel supplied, we found out that The Bat! for some ideotic reason left over 38000 (Thirty-eight-thousand) temp files in the temp directory. 0 bytes big, so no overhead there, but the plugin could not find any nonexisting temp file names to create...
After deleting all those files, everything got back to normal again.
Thanks Pavel! (And all the other guys involved off-course)
Peter