I'd love for someone to come and "remove" all of our "grass" too! Smothering can only happen as fast as I can find giant sheets of cardboard or other suitable materials.
A hard barrier like landscape timbers or bricks should be enough to stop Bermuda, just trim often enough so it doesn't go over. Severing all around the edge will help make sure nothing's still connected to plants outside the garden area, then pull/dig up everything within the garden, making sure to get all of the roots. Mow with the chute aiming away from beds so it doesn't throw grass seed into them. Hopefully you don't have torpedo grass!
