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
Was magic ring ever explained on a technical level? I thought all we know is it wants to be with sauron and it makes angels shit themselves.
For all we know putting it on a mouse gives everyone mouse nightmares and make them worship the mouse as mouse king before they take it straight to sauron.