You are focussing on the wrong problem.
The copy tool you need is simply `cp` and it's included in every distro. Any tool, whether it is `cp` or anything else, will have to mount the source and destination disks.
So your only problem is how to mount both the LVM source partition and the destination partition. There are 3 ways to do it:
-1- mount the LVM partition in Debian: that's easy, provided you use kernel 2.6 in Debian
-2- mount the Debian partition in Fedora: I don't know what file system you use in Debian, but if it's ext2/ext3 that should be no problem at all, easier than solution -1-
-3- use a live CD like RIP which allows you to mount both; if your live CD uses captive you can even copy to a Windows disk, though I wouldn't recommend it because you loose some information (uid, gid, permissions).
So I'd suggest you try solution -2- first, solution -1- second and if both of them fail, go and look for a live CD based on kernel 2.6 which has LVM support.