854
But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
I suspect it's more a question of proximity, the one not carrying it is much further and thus much less affected by it
If you carry the mouse, then it's almost as if you're carrying it yourself
They should have carried an eagle with the ring attached to him.
so just tie it to a stick and hold it in front of yourself