We study a variant of classical clustering formulations in the context of algorithmic fairness, known as diversity-aware clustering. In this variant we are given a collection of facility subsets, and a solution must contain at least a specified number of facilities from each subset while simultaneously minimizing the clustering objective ($k$-median or $k$-means). We investigate the fixed-parameter tractability of these problems and show several negative hardness and inapproximability results, even when we afford exponential running time with respect to some parameters. Motivated by these results we identify natural parameters of the problem, and present fixed-parameter approximation algorithms with approximation ratios $\big(1 + \frac{2}{e} +ε\big)$ and $\big(1 + \frac{8}{e}+ ε\big)$ for diversity-aware $k$-median and diversity-aware $k$-means respectively, and argue that these ratios are essentially tight assuming the gap-exponential time hypothesis. We also present a simple and more practical bicriteria approximation algorithm with better running time bounds. We finally propose efficient and practical heuristics. We evaluate the scalability and effectiveness of our methods in a wide variety of rigorously conducted experiments, on both real and synthetic data.