In Remark 2.3.2 of Emily Riehl's Categorical Homotopy theory, she says "it is not always possible to construct functorial deformations into chain complexes of projectives or cochain complexes of injectives. In the absence of functorial deformations, this construction defines a total derived functor, but not a point-set level functor."
She defines a total derived functor to be a Kan extension, and a point-set level derived functor is one which is a pointwise Kan extension. Here's how I interpret her remark in a more general setting:
Given homotopical categories $M$ and $N$, denoted by $\mathrm{ho}_M: M \to \mathrm{Ho}(M)$ and $\mathrm{ho}_N: N \to \mathrm{Ho}(N)$ the projections to the derived categories. Suppose we have a full subcategory $i: J \hookrightarrow M$ whose inclusion is a homotopical functor, such that every object in $M$ is quasi-isomorphic to an object in $J$. Then $i$ descends to an equivalence of derived categories $\mathrm{Ho}(J) \cong \mathrm{Ho}(M)$. Moreover, if $F: M \to N$ is a functor such that $F \circ i$ is homotopical, then $\mathrm{ho}_N \circ F \circ i$ descends to the derived functor $\mathbf{R}F: \mathrm{Ho}(M) \to \mathrm{Ho}(N).$
I have not checked that $i$ induces an equivalence of derived categories under these assumptions, but it certainly seems plausible to me. It suffices to show that any fully faithful homotopical functor descends to a fully faithful functor between derived categories, since $i$ obviously descends to an essentially surjective functor.
With regards to showing that this is composite I have defined is indeed the total right derived functor, the biggest issue is that I don't see how to get the unit natural transformation for the left Kan extension. When Emily Riehl constructed point-set derived functors via functorial deformations the natural transformation was inherited from the functorial deformation, but here I would have to construct it from scratch.
Anyway, is my conjecture correct? If so, does anyone know of a proof?