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Compound is the name horologists use, which is for them a pendulum of great interest.** It is a idealized form of the physical pendulum whose equation of motion is solved in intro. texts and the period is found by horologists.
John Haine*** uses the torques/moments of inertia method (which you used, and, I presume, do all the intro texts) to find the equation of motion (small angle approx.) and from that the period.
The problem you have w/ the Newtonian method illustrates the advantage of the Euler-Lagrangian formulation, which I suspect one can use. It ignores forces, only involving the KE and U (potential energy) as a function of generalized coordinates.