It is challenging to find optimum kinematic designs for non-standard robotic manipulators, e.g., medical, nuclear, and space manipulators, which are demanded to adapt to arbitrary complex tasks in constraints. Such design optimization can be modelled as a multi-dimensional non-convex optimization problem with nonlinear constrained conditions. However, it is non-trivial to ensure the essential reachability condition, i.e., the existence of continuous trajectories between demand positions for serial articulated manipulators, given complex spatial constraints, like obstacles and boundaries. Traditional solutions integrate standard motion planning or inverse kinematics algorithms within a kinematic-design optimization process, resulting in significant demand for time and computing resources. To accelerate design optimization at improved efficiency, we design a novel robust design framework built on a new kinematic design synthesis, which allows for simultaneously optimizing dimension and topology of a serial manipulator's kinematics for arbitrary tasks in constrained environments, using a generalised parametric kinematic model. Significantly, in contrast to standard solutions, we develop a novel computationally effective reachability verification method, which rapidly aborts infeasible motions by exploiting efficient collision checks, based on the Rapidly-exploring Random Tree (RRT) algorithm. The effectiveness of the proposed design framework is verified and evaluated by comparing to baseline benchmarks. Results demonstrate the novel design framework can accelerate kinematic design optimization by an order of magnitude compared to the current state-of-the-art, and optimise link dimension and joint type simultaneously of serial robots for cluttered environments.