Evaluating large language models typically requires thousands of benchmark items, making the process expensive and increasingly impractical. Existing protocols rely on average accuracy over fixed item sets, treating all items as equally informative—yet many items are too easy or too hard to distinguish between models, making evaluation both wasteful and imprecise. Item Response Theory (IRT) with Computerized Adaptive Testing (CAT) offers a principled alternative—separately estimating model ability and item difficulty, then adaptively selecting the most informative items. However, practical deployment is bottlenecked by costly item calibration, and standard greedy selection causes severe over-exposure of frequently picked items while leaving the rest of the bank under-utilized, which risks data contamination over time. We propose FAIR-Eval with two components: Feature-Driven Calibration (FDC), which jointly learns item parameters from text features and sparse responses, enabling low-cost bank expansion requiring only minimal model responses; and Maximum Fisher Information Plus (MFI+), which counteracts greedy selection by incorporating exposure control, while simultaneously enforcing difficulty coverage and domain diversity to systematically activate under-utilized items. On a 6,796-item bank (193 calibration models, 50 held-out), FAIR-Eval achieves Pearson r = 0.9882 and Spearman 𝜌 = 0.9828 against full evaluation using around 3% of items. Crucially, it broadens difficulty coverage from 60.0% to 68.7%, achieving a more comprehensive and robust assessment of model capabilities with minimal test length.