Enabling robot autonomy in complex environments for mission critical application requires robust state estimation. Particularly under conditions where the exteroceptive sensors, which the navigation depends on, can be degraded by environmental challenges thus, leading to mission failure. It is precisely in such challenges where the potential for FMCW radar sensors is highlighted: as a complementary exteroceptive sensing modality with direct velocity measuring capabilities. In this work we integrate radial speed measurements from a FMCW radar sensor, using a radial speed factor, to provide linear velocity updates into a sliding-window state estimator for fusion with LiDAR pose and IMU measurements. We demonstrate that this augmentation increases the robustness of the state estimator to challenging conditions present in the environment and the negative effects they can pose to vulnerable exteroceptive modalities. The proposed method is extensively evaluated using robotic field experiments conducted using an autonomous, full-scale, off-road vehicle operating at high-speeds (~12 m/s) in complex desert environments. Furthermore, the robustness of the approach is demonstrated for cases of both simulated and real-world degradation of the LiDAR odometry performance along with comparison against state-of-the-art methods for radar-inertial odometry on public datasets.