Stochastic reaction networks (SRNs) model stochastic effects for various applications, including intracellular chemical or biological processes and epidemiology. A typical challenge in practical problems modeled by SRNs is that only a few state variables can be dynamically observed. Given the measurement trajectories, one can estimate the conditional probability distribution of unobserved (hidden) state variables by solving a stochastic filtering problem. In this setting, the conditional distribution evolves over time according to an extensive or potentially infinite-dimensional system of coupled ordinary differential equations with jumps, known as the filtering equation. The current numerical filtering techniques, such as the filtered finite state projection (D'Ambrosio et al., 2022), are hindered by the curse of dimensionality, significantly affecting their computational performance. To address these limitations, we propose to use a dimensionality reduction technique based on the Markovian projection (MP), initially introduced for forward problems (Ben Hammouda et al., 2024). In this work, we explore how to adapt the existing MP approach to the filtering problem and introduce a novel version of the MP, the Filtered MP, that guarantees the consistency of the resulting estimator. The novel consistent MP filter employs a reduced-variance particle filter for estimating the jump intensities of the projected model and solves the filtering equations in a low-dimensional space. The analysis and empirical results highlight the superior computational efficiency of projection methods compared to the existing filtered finite state projection in the large dimensional setting.