We're on a rather fast san here (48 1TB drives 15 K rpm.) It would take over 100 minutes to copy 100 GB.

One way to do it is to move the full backup whenever you have time. Then take a diff backup (will be 1-5 GB at the most). Restore those 2 files. Then take a tail log backup and finish the restore with it.

That can also give you extremely small downtime while you do the switch.