Lie group variational integrators have the advantages of both variational and Lie group integrators, which preserve the momentum, symplectic form, holonomic constraints and the Lie group structure. In addition, their long-time energy stable behaviour and coordinate-independent nature make it quite suitable to simulate a variety of mechanical systems. The structure-preservation of a Lie group variational integrator implies its linearization is structure-preserving as well, thus we call such a linearization structured linearization. However, due to the implicit nature of variational integrators and the non-trivial differential structure of Lie groups, the structured linearization of Lie group variational integrators is much more complicated than that in generalized coordinates. In this paper, we formulate the structured linearization of Lie group variational integrators to synthesize existing analysis and control tools. To illustrate the utility of the paper, LQR controllers are constructed directly on constrained Lie groups for the asymmetric 3D pendulum and quadrotor with a suspended load, simulation results show that both controllers have a large basin of attraction.