The atmospheric composition of exoplanets with masses between 2 and 10 M$_oplus$ is poorly understood. In that regard, the sub-Neptune K2-18b, which is subject to Earth-like stellar irradiation, offers a valuable opportunity for the characterisation of such atmospheres. Previous analyses of its transmission spectrum from the Kepler, Hubble (HST), and Spitzer space telescopes data using both retrieval algorithms and forward-modelling suggest the presence of H$_2$O and an H$_2$--He atmosphere, but have not detected other gases, such as CH$_4$. We present simulations of the atmosphere of K2-18 b using Exo-REM, our self-consistent 1D radiative-equilibrium model, using a large grid of atmospheric parameters to infer constraints on its chemical composition. We show that our simulations favour atmospheric metallicities between 40 and 500 times solar and indicate, in some cases, the formation of H$_2$O-ice clouds, but not liquid H$_2$O clouds. We also confirm the findings of our previous study, which showed that CH$_4$ absorption features nominally dominate the transmission spectrum in the HST spectral range. We compare our results with results from retrieval algorithms and find that the H$_2$O-dominated spectrum interpretation is either due to the omission of CH$_4$ absorptions or a strong overfitting of the data. Finally, we investigated different scenarios that would allow for a CH$_4$-depleted atmosphere. We were able to fit the data to those scenarios, finding, however, that it is very unlikely for K2-18b to have a high internal temperature. A low C/O ratio ($approx$ 0.01--0.1) allows for H$_2$O to dominate the transmission spectrum and can fit the data but so far, this set-up lacks a physical explanation. Simulations with a C/O ratio $<$ 0.01 are not able to fit the data satisfactorily.
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