864
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
The mouse would have chewed through the pocket and run away as soon as it could
Or possibly climb the nearest thing to get eaten by a bird, kinda like what Frodo did with the nazgul.
Frodo ate a Nazgûl?
They're surprisingly low calorie.