I just installed my rear ARB tire carrier. Initially I had the rattle issue as well, but made a few adjustments and got rid of it.
1) make sure to tighten the rod that connects the tire swing to the tailgate bracket so that the silver tube fits firmly in the nylon block when closed. I had to readjust mine at the end by almost a half inch.
2) make sure the rubber bumpstops or "tyre rests" as the are called in the instructions are pressed firmly against the tailgate when closed after you do step one.
3) make sure to drill the hole in the wheel mount receiver and put the 3rd bolt in. My tire was rattling because the 2 side bolts were not enough to keep the wheel mount snug. (it had up and down play with the side bolts only, the vertical bolt will eliminate this) also make sure to push the tire in as tight as possible before you mark the hole.
Hope that helps, although I may be stating the obvious here. It sure was a pain to get on. Good luck.