@ConnorMalin For sure, it is. Yet, I had the impression that the construction of $Q$ was some kind of localization at admissible epi (not exactly though). I was expecting that localizing further would lead to a groupoid, then to a fibrant simplicial set, with no need of any fibrant replacement.

@PhilTosteson Thanks! Hmm... a very naive interrogation: if one is going to take a fibrant replacement after all, why bother with the intermediate step $Q(\mathcal{E})$?

This is enlightening, thank you very much for your answer! And your computations, as well

Let me precise: let $F:A\to B$ be a left-exact functor of abelian categories, and let $Q:K(A)\to D(A)$ be the localization functor (resp. $K(B)\to D(B)$). We say the pair $(RF,\varepsilon)$ - where $RF:D(A)\to D(B)$ is a functor and $\varepsilon:Q\circ F\to RF\circ Q$ a natural transformation - is a right derived functor of $F$ if, for each pair $(G,\eta)$ as above, there exists a unique natural transformation $\alpha:RF\to G$ such that $\eta=\alpha\circ \varepsilon$. Is your complex $C_{\mathcal{E}}(X,-)$ - or rather its shifted cone? - a right derived functor of $Hom_{\mathcal{E}}(X,-)$?

Dear @Leonid, thank you for your relevant answer! Seems indeed that I am in possession of the complex $C_{\mathcal{E}}$ you are describing there. Something that I do not quite get is how your construction recovers $RHom_{\mathcal{E}}(Y,-)$ whenever the latter exists, and whether it can be used to prove that $RHom_{\mathcal{E}}(Y,-)$ indeed exists. When I refer to $RHom_{\mathcal{E}}(Y,-)$, I mean the the unique functor satisfying the "universal property of the right derived functor of $Hom_{\mathcal{E}}(Y,-)$". Do you have some thoughts on that?

In practice I am struggling with a more involved category without enough projectives. Here I am looking at the "baby-case" of modules, and that is the reason why I am trying to avoid projective resolutions. But thanks, your reference is very relevant.

Thank you! It took me some time to complete the argument, but now it is good :)