Actually, after stepping away from the computer for a little while, I've realized that the question I asked at the end of my above comment is a bit stupid, as the good properties come directly from the fact that one is using Hom, along with some acyclicity thing, so a "generalization" is not really meaningful. In my defense, it's 6 in the morning where I am now (and I haven't slept yet...). Thinking about this has actually straightened a few things out in my mind, I think, so I will accept the answer on that basis.

While this isn't the kind of thing I was hoping for, it is nonetheless interesting and I'm glad to have read it (though I was aware of some of these things already, just missing some perspective I think). I'm unsure if I should mark it as an accepted answer because though it doesn't answer what I'm after, I do quite like what you've written. Perhaps a question: defining local objects (or their dual) is based on behaviour under the action of some Hom functor. Is there a useful generalization of this apparatus to other functors? (One cannot guarantee that K-projectives exist, hence why I ask.)

@ZhenLin In other words, the data written in 2? My question was basically under what conditions one could upgrade such a thing to a functorial choice. I'm also aware that e.g. assuming the weak equivalences form a right/left multiplicative system, one can follow e.g. Kashiwara–Schapira to construct one's desired absolute Kan extensions (this is all written in the original question). My take-away is that there is no way of doing what I want, then?

@ZhenLin Well, the reason I'm "fixating" on them is because I like the proofs in that context and specifically want to know how practical it is to exclusively use them because of that. I don't think this is so strange. With regards to the last thing you said, out of curiosity, what are you referring to? Just that one can construct derived functors in the context of model categories, or something else?

@ZhenLin Perhaps, but on the other hand, if it works then it works (and I don't see anything that would prevent it from working). It's also somewhat besides the point, because as I said, if there is a way to do what I want on the level of $\mathbf{Ch}(\mathcal{A})$ (even better: in some more arbitrary homotopical category) then that would leave me very satisfied. To be as clear as I can about what I want: fix some $\mathcal{C}'\subseteq\mathcal{C}$ as in 1, and assume that one can do 2 as stated (i.e. non-functorially). I want a way to guarantee that 2 can be upgraded to a functorial choice.

@ZhenLin I'm satisfied with "anything" which can give functoriality, in the sense that I'll take what I can get. If something better exists, then I'd rather have that (for example, while the applications I care about are in contexts like $\mathbf{Ch}(\mathcal{A})$ and $\mathbf{K}(\mathcal{A})$, I'd prefer methods that work in broader contexts as well). As an additional note: if one can get full model structures with functorial factorizations, that's nice, but I also do not demand it at all (I only want the deformation part that could, for example, come from such a thing).

@LeoAlonso Are you sure you mean section 2.9? In both the copies I have, it only goes up to 2.7. Though, on a separate note, I don't think I remember Lipman discussing anything about functoriality, and since I'm interested in the approach using deformations, functoriality is important. If one excludes functoriality, then I think all my questions are answered by the contents of Kashiwara–Schapira. (Maybe that also clarifies what I want? I don't know)

@DenisT To be honest, I don't really have any particularly exotic things in mind, and the already mentioned methods are perfectly fine (e.g. that of Spaltenstein). The linked answer on K-flat resolutions is a good example of the kind of situation I'm talking about (where one is interested in getting the derived tensor product). Mainly, I just find the explanation there a bit unsatisfying, and was hoping there was something which didn't rely on such an explicit situation, if that makes sense.

@LeoAlonso What I mean is that I can't start with some random/arbitrary choice for $\mathcal{C}'$. There are methods that work when you have projectives, or K-projectives, etc., but what I'm asking for is something that will work more generically. I'm prepared for such a thing not to exist, but I have a wish that it does.

I made an edit to add some more context/explanation, which also hopefully shows what kind of unreasonable thing I'm really asking for (but still hoping is possible in restricted but practically applicable scenarios).

The problem I have with the first part is that it assumes one has a model structure, and that one is working with functors suitably compatible with that model structure. In practice, this can be really troublesome to arrange for, regardless of whether one wants the model structure to have functorial factorizations (but on the other hand, it seems hard to arrange for anything).

Preferably, I'd want situations in which one can ensure that 1 and 2 can be done functorially, to the extent that one can realistically expect that to be possible. This is mostly because I'm interested in seeing what can be done if one is forced to use only the deformations approach; in reality, one can of course use something like Kashiwara–Schapira's results to sidestep this need entirely.