Lie groups offer an elegant, efficient, and geometrically exact formulation for modelling highly flexible structures. In particular, the left-trivialized description of the nodal velocities allows for a strain-objective, local description of the beam on the Lie algebra [1]. This property is also preserved under finite element discretization, allowing for a more accurate representation of the dynamics through the explicit coupling of rotations and translations. To date, only the shooting algorithm [2] has been employed for computing Nonlinear Normal Modes (NNM’s) of SE(3) beams. However, the shooting method has several drawbacks, including the production of extraneous higher-order modal interactions. As opposed to shooting, one can manually select and truncate the relevant frequency content of the response using the Harmonic Balance Method (HBM). In this paper, we derive the HBM equations applied to the nonlinear local SE(3) Lie group model to compute the NNM’s and forced response curves via pseudo-arclength numerical continuation. An efficient implementation, achieved through careful sampling, is used to reduce computational load. This idea is propagated to Hill’s method for stability analysis, where the trigonometric collocation Alternating Frequency-Time (AFT) method is adapted for an efficient Galerkin projection [3] for the nonlinear terms. A trust-region Newton method is applied to solve the equations of a nonlinear cantilever beam and a doubly clamped beam, revealing strongly nonlinear features at higher deflection levels. The results show loops with a strong third harmonic in the doubly clamped beam. The results were validated against the shooting method, showing good scalability.