Fair clustering has recently received considerable attention where numerous distinct fairness notions are developed. Despite being well-justified, these fairness notions are frequently studied in isolation, leaving the need to explore how they can be combined. Building on prior work, we focus on the doubly constrained fair clustering that incorporates two widely adopted demographic representation fairness notions in clustering: group fairness and data summarization fairness. Both fairness notions extend classical clustering formulation by associating each data point with a demographic label, where group fairness requires each cluster to proportionally reflect the population-level distribution of demographic groups, and data summarization fairness ensures the chosen facilities maintaining the population-level demographic representation of each group. In this paper, we study the Fixed-Parameter Tractable (FPT) approximation algorithms for doubly constrained fair clustering under the k-median objective, referred to Df-k-Med. The previous algorithms typically enumerate different demographic groups or construct fairness coreset, parameterized by both the number of opened facilities and demographic labels. By further leveraging the local fairness information, we propose a color-agnostic structural method that obtains the parameterized result independent of the number of demographic labels while effectively handling the combination of both fairness constraints. Specifically, we design a constant factor approximation for the Df-k-Med problem with fairness violation by one, which runs in FPT(k)-time, where k is the number of opened facilities.