We study \( G_{2} \) -manifolds obtained from circle bundles over symplectic \( \operatorname {SU}(3) \) -manifolds with \( T^{2} \) -symmetry. When the geometry is multi-Hamiltonian, we show how the compact part of the resulting multi-moment graph for the \( G_{2} \) -structure may obtained cohomologically from the base. The lifting procedure is illustrated in the context of toric geometry.