If you think the rattle is the internal brace not securely glued to the sides then you must try to re-glue that brace and screw into it through the side panels. If you don't, then the brace is ineffective. It should reduce movement of the side panel both inwards and outwards.
You cannot eliminate panels movement, but effective bracing that creates different sized small unsupported panels can go a long way to reducing the "box" noise to near inaudible.
