Statistical modeling of claim severity distributions is essential in insurance and risk management, where achieving a balance between robustness and efficiency in parameter estimation is critical against model contaminations. Two \( L \)-estimators, the method of trimmed moments (MTM) and the method of winsorized moments (MWM), are commonly used in the literature, but they are constrained by rigid weighting schemes that either discard or uniformly down-weight extreme observations, limiting their customized adaptability. This paper proposes a flexible robust \( L \)-estimation framework weighted by Kumaraswamy densities, offering smoothly varying observation-specific weights that preserve valuable information while improving robustness and efficiency. The framework is developed for parametric claim severity models, including Pareto, lognormal, and Fr{é}chet distributions, with theoretical justifications on asymptotic normality and variance-covariance structures. Through simulations and application to a U.S. indemnity loss dataset, the proposed method demonstrates superior performance over MTM, MWM, and MLE approaches, particularly in handling outliers and heavy-tailed distributions, making it a flexible and reliable alternative for loss severity modeling.