Treating morphology and control as separate design problems has constrained the efficiency and adaptability of quadrupedal robots, while current co-design approaches are often constrained by restricted design spaces. We overcome this limit by introducing Dogger, a bio-inspired quadruped with modular three-segment legs, four-bar linkages and adjustable passive knee compliance. A two-stage framework combining reinforcement learning and Bayesian optimization is developed: the morphology-conditioned control policy is trained across designs and then held fixed during task-specific design selection. The resulting designs and policies deployed zero-shot reduce the cost of transport by 18.6% and 35.7% for inspection and payload-carrying tasks. Comparisons demonstrate that full geometry-compliance co-design outperforms fixed and restricted design scopes. Cross-task tests further reveal distinct task-specific morphologies with shared mechanical trade-offs. These results provide hardware-level evidence that expanding the physical design space alongside learned control is essential to realize the full potential of task-driven morphology-control co-design in legged robotics.

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