Methods I've seen are either brackets into the false floor panel, or straps. Brackets are a pain if the spare ever has to come out, and the false floor is flimsy - straps are at least more user friendly but people often do them a bit half assed and make it look like they're transporting a box on the back of a ute.

I'd look at attaching a couple flat webbed straps with clips to the back of the box, and then attach the other ends of the straps to the body to existing fasteners; one up the top behind the interior panel, and one under the boot carpet. Box then clips to straps to keep it in place, and it's easily removable should you need to get it out.