Developing natural and diverse locomotion controllers for quadruped robots that can adapt to complex terrains while preserving motion style remains a significant challenge. Existing imitation-based methods face a fundamental optimization trade-off: strict adherence to motion capture (mocap) references penalizes the geometric deviations required for terrain adaptability, whereas terrain-centric policies often compromise stylistic fidelity. We introduce LatentMimic, a novel locomotion learning framework that decouples stylistic fidelity from geometric constraints. By minimizing the marginal latent divergence between the policy's state-action distribution and a learned mocap prior, our approach provides a conditional relaxation of rigid pose-tracking objectives. This formulation preserves gait topology while permitting independent end-effector adaptations for irregular terrains. We further introduce a terrain adaptation module with a dynamic replay buffer to resolve the policy's distribution shifts across different terrains. We validate our method across four locomotion styles and four terrains, demonstrating that LatentMimic enables effective terrain-adaptive locomotion, achieving higher terrain traversal success rates than state-of-the-art motion-tracking methods while maintaining high stylistic fidelity.