Can you just raise it? I actually would consider moving it, do you want to be reaching into cactus or digging up the bed if you have to do repairs? It's fairly easy to move them and since you are going to have that area dug up do it now.
Tally HO!

This is the front/side of the garden and I intend to hide some of it with a sotol and perhaps some small boulders. I won't have to climb through cactus. I can't move the box.

ah, just buy a bigger box, they make them in different sizes but they can be hard to find. You can make your own "extension" by using another box. Cut it down as needed.
Good luck, show us pics when you get cactus in.
Tally HO!