This paper investigates a fluid reconfigurable intelligent surface (FRIS)-assisted Rydberg Atomic REceiver (RARE) architecture under magnitude-only heterodyne readout. We show that, unlike conventional coherent systems, the optimal propagation environment is fundamentally governed by the receiver's nonlinear measurement structure. In particular, under the strong-reference regime, symbol detection is limited by residual quadrature leakage after reference alignment, motivating a receiver-induced channel shaping approach rather than conventional channel-centric optimization. Based on this insight, we formulate a signal-independent leakage minimization problem that jointly optimizes the FRIS port set, finite-resolution phase shifts, and the transmit beamformer, resulting in a nonconvex mixed discrete-continuous design. To address this, we develop an alternating-optimization (AO) framework comprising: (i) a closed-form eigenvector solution for widely-linear beamforming, (ii) cross-entropy method (CEM)-based combinatorial port selection, and (iii) coordinate-descent (CD) phase refinement with guaranteed monotonic descent. Simulation results demonstrate fast convergence and consistent bit-error-rate (BER) gains across various modulation orders and receiver dimensions. Moreover, the proposed FRIS-enabled design achieves near-exhaustive performance with significantly reduced complexity and consistently outperforms conventional RIS schemes with fixed elements, highlighting the effectiveness of spatial reconfiguration in suppressing quadrature leakage and the additional spatial degree-of-freedom (DoF) enabled by FRIS for reliable atomic-MIMO detection.