Turning a simple polygon with holes into exterior-bounded only
The closest vertices between the inner and outer polygons do not necessarily produce a valid solution as shown in the following figure:



You just need to find a pair of vertices between the inner and outer polygons whose corresponding edge does not intersect either polygon.
Search for vertices $u$ and $v$ from the inner and outer polygons such that the edge $(u,v)$ does not cross any other edge (some care needs to be taken that you don't
slip between two edges as shown on the right above -- choosing the closer vertex avoids this problem).

A straight forward approach would scan all such $u,$ $v,$ and all edges
resulting in $O(n^3)$ time in the worst case -- in practice you'll find a solution fairly quickly. This can be sped up by placing all the vertices in a BSP tree whose half-spaces are defined by the polygon edges -- then you only considering vertices $v$ in the same half-space as $u.$