Solution:

A disphenoid exists (as the tetrahedron formed by the face-plane midpoints of a box) exactly when the common face triangle is acute, and its total surface area is $4$ times one face's area. We want the smallest-area acute scalene integer triangle. The candidates $(2,3,4)$ and $(3,5,6)$ are obtuse, and $(3,4,5)$ is right (giving a degenerate flat figure), but $(4,5,6)$ is acute since $4^2+5^2\gt6^2.$ By Heron with $s=\tfrac{15}2,$ its area is $\sqrt{\tfrac{15}2\cdot\tfrac72\cdot\tfrac52\cdot\tfrac32}=\tfrac{15\sqrt7}{4}.$ The total surface area is $4\cdot\tfrac{15\sqrt7}{4}=15\sqrt7.$ Thus, the correct answer is D.