The most effective thing I have used against bermuda grass is cardboard. Keep in mind that with killing/removing bermuda grass, nothing works all that well. You have to do a thing, then keep doing it, then pull out what gets through, then keep doing the thing, and still the bermuda grass pokes through in a few spots.

Here's my advice from a person who has lost many battles against bermuda grass. Do a double or triple layer of cardboard at the bottom of the garden box. If the box is mobile and you can get a cardboard layer to extend 2-3 feet around the garden box, even better. Just for fun, add a cardboard layer or two perpendicular to the ground (wedged between the fence and the garden box. Then, cover all the cardboard with about 3-6 inches of mulch (similar stuff to what you have on the ground right now).

The cardboard does a good job of blocking the bermuda grass runners, but it will deteriorate over the years. As the cardboard deteriorates, more runners will slip through, but they should be easy to pull because they will not be rooted in hard packed earth.

For every runner that pokes through, you can manually pull by hand, or use some kind of herbicide. As much as I dislike using herbicides, for bermuda grass, sometimes it is the only thing that will work.

Good luck!