So my question is, is there a way to preload all unicode properties so that I don't have to worry about this?
Probably, but I wouldn't look to figure out what it is, I would install perl/modules/everything-you-need under the chroot jail, so it works like regular perl. See links for chroot setup
Or at least a way to get a clean failure when it can't find one of these properties, instead of mysterious misbehavior?
Well, AFAIK, even the buggy perl-5.10.x ought to give an error when a particular needed unicore file is missing, so you could try upgrading? Or writing a minimal testcase and submitting it using perlbug?