How would I algorithmically "stretch" polygons on a plane by re-scaling the distances between interior points?
You might try using the spring model: https://en.wikipedia.org/wiki/Force-directed_graph_drawing.  Or, you could use optimization methods.  Build an objective function $\Psi$ that is a sum of penalty terms.  For each pair of mapped points $a^*,b^*$ that you are hoping will be at distance $c_{a^*,b^*}$, you have a penalty term $(d(a^*,b^*) - c_{a^*,b^*})^2$.  Also, for each point $a^*$, add a penalty term that is zero throughout most of the interior of $A$, but increases when you get close to the boundary of $A$ and rises to $+\infty$ for all points outside $A$.  Finally, find an arrangement of points that minimizes this objective function.  You could then apply any of a number of optimization methods to do this, such as gradient descent or others.  Stress majorization may be useful.