Figure 1.

The orange curve (L) represents the exact potential given by (3) divided by $A=G{m}^2/r_2$. It is finite at the origin! (no UV troubles at all then). The violet curve (H) represents the long-range (Newtonian) approximate potential $V_3/A$ given by (6), where the horizontal coordinate $x=r/r_2$ and $r_2={(a/b)}^{1/3}$.