It's super interesting to see your counter-example, especially when the following is true:
If $R$ is a Gorenstein domain of dimension 4, then finite reflexive modules are projective.
Proof: Since both are local properties, we assume $(R,m)$ is local. As Greg pointed out, $\operatorname{Ext}_R^1(M,R) = \operatorname{Ext}^2_R(M,R) = 0$. By local duality, this says $H^3_m(M) = H^2_m(M) = 0$. We need to show that $\operatorname{Ext}^i_R(M,R)$ vanish for all $i = 3,4 $, iff the lower local cohomologies vanish, i.e depth $M \ge 2$. But this is clear since $R$ has depth $4\ge 2$, any $\operatorname{Hom}_R(N,R)$ will have depth $\ge 2$ (see https://stacks.math.columbia.edu/tag/0AV5).