Juust to indicate that there is a solution to the problem:
the trivial solution for the smallest graph with $k$ edge-disjoint Hamilton paths between a given pair of vertices is to take a graph $G'$ with $k$ edge-disjoint Hamilton cycles and split one of its vertices.

The smallest graph with two edge-disjoint Hamilton cycles is $K_5$ and so the smallest graph that solves the original problem is a vertex-split $K_5$ depicted below:    

[![enter image description here][1]][1]


  [1]: https://i.sstatic.net/uJmnE.png