  We recently described a protocol for computing the potential energy in
heterotic M-theory for the dilaton, complex structure and K\"ahler moduli. This
included the leading order non-perturbative contributions to the complex
structure, gaugino condensation and worldsheet instantons assuming a hidden
sector that contains an anomalous U(1) structure group embedded in $E_8$. In
this paper, we elucidate, in detail, the mathematical and computational methods
required to utilize this protocol. These methods are then applied to a
realistic heterotic M-theory model, the $B-L$ MSSM, whose observable sector is
consistent with all particle physics requirements. Within this context, it is
shown that the dilaton and universal moduli can be completely stabilized at
values compatible with every phenomenological and mathematical constraint -- as
well as with $\Lambda$CDM cosmology. We also show that the heterotic M-theory
vacua are consistent with all well-supported Swampland conjectures based on
considerations of string theory and quantum gravity, and we discuss the
implications of dark energy theorems for compactified theories.
