Here is something I did involving LVM, that may give you ideas on how to proceed:http://mandrivausers.org/index.php?/topic/83474-remotely-grow-and-shrink-lvm-partitions-mandriva
The problem you have with moving /var seems similar to the problem I had moving /usr under LVM. In short (if you’re not interested in reading the above link), the steps are:
— copy the data to the new location,
— change fstab so that the new device node is mounted instead of the old one,
— lazy-umount the old location and quickly mount the new one (mount -a), then reboot.
I wrote this on-the-fly and did not test. Make sure you understand and feel confident before trying this.