Transferring articulated motion from monocular videos to rigged 3D characters is challenging due to pose ambiguity in 2D observations and morphological differences between source and target. Existing approaches often follow a reconstruct-then-retarget paradigm, tying transfer quality to intermediate 3D reconstruction and limiting applicability to categories with parametric templates. We propose MorphGS, a framework that formulates motion retargeting as a target-driven analysis-by-synthesis problem, directly optimizing target morphology and pose through image-space supervision. A rig-coupled morphology parameterization factorizes character identity from time-varying joint rotations, while dense 2D-3D correspondences and synthesized views provide complementary structural and multi-view guidance. Experiments on synthetic benchmarks and in-the-wild videos show consistent improvements over baselines.