Crape myrtles root very easily for us in Fladeedah. Years ago I made a temporary fence with crape myrtle shoots that had been cut down the week before, and they all rooted in place. This is one shrub that I "transplant" by ripping it out of the ground and tossing it in a ditch or hole, then kicking some dirt back over it. Not the recommended course of action. Jes sayin'.
Cut down with a spade as close to your fence as possible. There will probably be a root section that your offset is growing from. You want a sizable amount of root, as opposed to a rootless cutting. Kinda whatever you get. Set it in your prepared hole and tamp your soil in around it, watering well. Mulch, and keep some shade over it for a few days minimum. If it's small enough, a lawn chair can shade it.
Expect some wilting of the existing greenery, and loss of any flowers or buds on this piece, but once you see new growth you are golden.