Background/Objectives: Neoadjuvant chemotherapy is the standard of care for patients with locally advanced gastric cancer. Inflammatory and nutritional indices have been proposed as potential prognostic biomarkers in this setting. This study aimed to evaluate their association with pathological response and relapse-free survival (RFS) in patients with locally advanced gastric cancer treated with the perioperative FLOT regimen. Methods: This multicenter retrospective study included 120 patients treated with perioperative FLOT between 2018 and 2022. Pathological response was assessed using the Becker regression grading system. Pretreatment inflammatory and nutritional biomarkers were calculated from baseline data. Association with pathological responses and RFS were analyzed using logistic regression, Kaplan–Meier estimates, and Cox models. Results: Pathological response was strongly associated with higher radiologic and R0 resection rates and lower recurrence (all p < 0.001). None of the biomarkers correlated significantly with pathological response. Pathological response was the strongest prognostic factor for RFS (p < 0.001), while age (p = 0.011) and histologic subtype (p = 0.004) were also independent predictors. The CEA/albumin ratio showed a trend toward significance (HR 1.805; 95% CI 0.918–3.551; p = 0.087), and patients with lower ratios had longer RFS (median not reached vs. 15.8 months, p = 0.014). Conclusions: Pathological response remains the most powerful prognostic factor in locally advanced gastric cancer treated with perioperative FLOT. Although inflammatory and nutritional indices alone may not predict treatment response, the CEA/albumin ratio demonstrates potential prognostic value for RFS. Larger prospective studies with standardized cut-off values are warranted to validate these findings and to further explore the dynamic prognostic role of immunonutritional markers.