865
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
This would create quite an evil mouse. I imagine the risk here is that the mouse would break free and run away with the ring and bring it to Sauron.
Edit: Someone already beat me to it.
You need legs to run away
it would take the cookie, and you know what happens when you give a mouse a cookie
Also good luck finding an invisible mouse.