In this talk, we discuss compact complex 3-dimensional non-K\"ahler Bismut Ricci flat pluriclosed Hermitian manifolds (BHE) via their dimensional reduction to a special K\"ahler geometry in complex dimension 2. The reduced geometry satisfies a 6th order non-linear PDE which has infinite dimensional momentum map interpretation, similar to the much studied K\"ahler metrics of constant scalar curvature (cscK). We use this to associate to the reduced manifold or orbifold Mabuchi and Calabi functionals, as well as to obtain obstructions for the existence of solutions in terms of the automorphism group, paralleling results by Futaki and Calabi--Lichnerowicz--Matsushima in the cscK case. Through the obstruction theorem, we show that the quotients of $SU(2)\times SU(2)$ or $SU(2)\times \mathbb{R}^3$ as the only non-K\"ahler BHE $3$-folds with $2$-dimensional Bott--Chern $(1,1)$-cohomology group, for which the reduced space is a smooth K\"ahler surface. Lastly, we discuss explicit solutions of the PDE on orthotoric K\"ahler orbifold surfaces which yield infinitely many non-K\"ahler BHE structures on $S^3\times S^3$  and $S^1\times S^2 \times S^3$.