Movable antennas (MAs) have recently emerged as a promising technology to enhance the wireless sensing performance by exploiting additional spatial degrees of freedom. In this work, we consider an MA-enabled near-field target localization problem under a device-based sensing architecture, where the sensing target actively transmits signals and the sensing receiver estimates its location based on the received signals. We adopt the squared position error bound (SPEB) as the sensing performance metric and aim to minimize the target's SPEB by optimizing the antenna placement at the sensing receiver. The resulting optimization problem is highly non-convex due to the minimum antenna spacing constraints and the intricate dependence of the SPEB on the antenna locations and the intractable form of SPEB. To tackle this problem, we first relax the inter-spacing constraints and accordingly formulate an antenna position distribution problem in terms of degree-4 truncated moments. The reformulated moment-based problem admits an equivalent semidefinite programming formulation and can be optimally solved using standard convex optimization solvers. Using truncated moment theory, we then recover an optimal antenna position distribution supported on a finite number of locations and construct an efficient clustered placement policy. Numerical results demonstrate that the constructed optimal distribution exhibits a three-cluster structure with source-dependent weights. The proposed antenna placement method outperforms conventional array geometries and closely approaches the exhaustive-search benchmark scheme.

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