This paper presents a comprehensive physical layer security (PLS) framework for fluid antenna system (FAS)-aided short-packet communications under the variable block-correlation model (VBCM). We consider a downlink wiretap scenario in which a base station transmits confidential short packets to a legitimate receiver user (RU) in the presence of an eavesdropper user (EU), where both the RU and EU are equipped with fluid antennas. Unlike existing FAS security analyses that rely on constant block-correlation models or infinite-blocklength assumptions, we incorporate the VBCM to accurately capture the non-uniform spatial correlation structure inherent in practical FAS deployments. By employing a piecewise linear approximation of the decoding error probability and Gauss-Chebyshev quadrature, we derive closed-form and asymptotic expressions for the average achievable secrecy throughput (AAST). We further prove that the AAST is monotonically non-decreasing in the number of RU ports, which reduces the three-dimensional joint optimization of transmit power, blocklength, and port number to a two-dimensional grid search (GS). Numerical results demonstrate that the FAS-aided system achieves up to an order-of-magnitude secrecy throughput improvement over conventional fixed-position antenna systems, and reveal that blocklength selection is the most critical design parameter in the joint optimization.