Large Language Models (LLMs) are increasingly being employed for downstream tasks such as tabular classification, where predictions can directly influence individuals in high-stakes domain such as hiring, education and criminal justice. As the adoption of LLMs for such decision making tasks continues to grow, ensuring fairness in their predictions becomes increasingly important. Existing approaches to ensure fairness in tabular data, typically process test instances independently, limiting the models awareness of the demographic composition and predictions of other instances, which is particularly important for reasoning about group-level fairness. To address this limitation, we propose FABLE (FAir Batch-based LLM framEwork) for tabular classification, a batch-based framework that jointly processes multiple test instances, enabling LLMs to reason over demographic information and prediction outcomes within a batch. To mitigate positional bias introduced by batch processing, the framework generates multiple shuffled versions of each batch and aggregates the resulting predictions through voting. We evaluate FABLE on three tabular classification datasets using three popular LLMs and four prompting configurations spanning zero-shot, few-shot, direct, and self-reflective prompting. Experimental results demonstrate that incorporating demographic information and explicit fairness constraints enable LLMs to achieve substantial fairness improvements across diverse datasets and fairness notions, highlighting their potential as fairness reasoners for tabular classification and promoting more equitable and accountable decision-making.