In a continuous-variable quantum key distribution (CV-QKD) protocol, which is based on heterodyne detection at the receiver, the application of a noiseless linear amplifier (NLA) on the received signal before the detection can be emulated by the postselection of the detection outcome. Such a postselection, which is also called a measurement-based NLA, requires a cutoff on the amplitude of the heterodyne-detection outcome to produce a normalizable filter function. Increasing the cutoff with respect to the received signals results in a more faithful emulation of the NLA and nearly Gaussian output statistics at the cost of discarding more data. While recent works have shown the benefits of postselection via an asymptotic security analysis, we undertake an investigation of such a postselection utilizing a composable security proof in the realistic finite-size regime, where this tradeoff is extremely relevant. We show that this form of postselection offers only a small fraction of the asymptotic improvement in the finite-size regime. This postselection can improve the secure range of a CV-QKD over lossy thermal channels if the finite block size is sufficiently large and the optimal value for the filter cutoff is typically in the non-Gaussian regime. The relatively modest improvement in the finite-size regime as compared to the asymptotic case highlights the need for new tools to prove the security of non-Gaussian cryptographic protocols. These results also represent a quantitative assessment of a measurement-based NLA with an entangled-state input in both the Gaussian and non-Gaussian regime.