actually copying the bytes between the two devices. At least the mv program (and thus, maybe also the rename() syscall?) on my Debian system does such copying nowadays...

It's a good thing OP is using File::Copy, since that actually does copy the bytes (and only when necessary, AFAIK) - I belive this is what the OS's mv does too. I just tried rename across filesystems and did in fact get the Invalid cross-device link error.