This paper presents a new two-step optimal fourth-order iterative scheme for computing roots of known multiplicity m ≥ 1 of a nonlinear equation. The method extends a Padé-type rational weight function, originally devised for simple roots, to the multiple-root setting through a single m-th-root substitution that scales naturally with the multiplicity and reduces exactly to its simple-root parent when m = 1. Using symbolically verified Taylor-series analysis, the scheme is shown to have fourth-order convergence for arbitrary m while requiring only three functional evaluations per iteration, thereby attaining equality in the Kung–Traub optimality bound. A closed-form asymptotic error equation is derived in terms of the multiplicity m, a single tunable real parameter α, and the normalized Taylor coefficients of f at the root, offering an explicit mechanism for minimizing the error constant without sacrificing convergence order. Numerical experiments on benchmark problems and physically motivated multiple-root problems confirm the predicted order and demonstrate the competitiveness of the proposed scheme against existing optimal fourth-order multiple-root solvers.