Compactifications of the heterotic string, to first order in the α′ expansion, on manifolds with integrable G2 structure give rise to three-dimensional = 1 supergravity theories that admit Minkowski and AdS ground states. As shown in, such vacua correspond to critical loci of a real superpotentialW. We perform a perturbative study around a supersymmetric vacuum of the theory, which confirms that the first order variation of the superpotential, δW, reproduces the BPS conditions for the system, and furthermore shows that δ2W = 0 gives the equations for infinitesimal moduli. This allows us to identify a nilpotent differential, and a symplectic pairing, which we use to construct a bicomplex, or a double complex, for the heterotic G2 system. Using this complex, we determine infinitesimal moduli and their obstructions in terms of related cohomology groups. Finally, by interpreting δ2W as an action, we compute the one-loop partition function of the heterotic G2 system and show it can be decomposed into a product of one-loop partition functions of Abelian and non-Abelian instanton gauge theories.