Ok, so after discussing with some Madoka/Hunger games fans, I think the easiest thing to have happen is that she has it on her, but it gives her no powers. And you are free to play with having it stolen/destroyed as long as she stays in the games, she would have it back again when she was brought back.

Because of the way death works, which I can't go into too many details of, she would have it refreshed every cycle.