You might find this helpful.

As for the comment of quotients vs. subs, I'm aware of the distinction in convention (i.e., do we use $\operatorname{Sym}\mathcal{E}$ or $\operatorname{Sym}\mathcal{E}^{\vee}$ when defining $\mathbb{P}(\mathcal{E})$)-- good to mention just in case someone isn't familiar, of course.

@Z.M oh, of course! That makes a lot of sense -- then we just need to check that the functor $A\mapsto (\mathsf{Mod}_A)_{/A^{n+1}}$ preserves the subcategories described. I suppose it's also fine to use the slice category over $A^{n+1}$ here since the pullback/base change along $A\to B$ will send $A^{n+1}$ to $B^{n+1}.$ However, I don't see why we need point 3.

Not an answer to your question, but something related you may or may not already be familiar with -- the category of $R$-algebras has an interesting 2-categorical structure where 1-morphisms are bimodules and 2-morphisms are the obvious notion of bimodule homomorphisms.

@DavidBenjaminLim If I assume that $X$ in addition is itself qcqs, then I believe the argument below making use of your observation from theorem 3.2 proves the result I want.

@DavidBenjaminLim I should have noticed that much sooner! However, I'm don't think that totally solves the issue, unless I'm missing something: I'm want to consider the relative diagonal of $f : X\to\mathcal{L}\!\operatorname{og}_{(S,\mathcal{M}_S)},$ not the diagonal of $\mathcal{L}\!\operatorname{og}_{(S,\mathcal{M}_S)}$ itself. After all, $\Delta_f$ is quasi-compact if and only if $f$ is quasi-separated.

@DavidBenjaminLim Yeah, I've been struggling with this as well -- thanks for your thoughts. The quasi-coherated derived pushforward is hard to work with, especially after the derived pullback. The assumption comes from the following: we can define Hochschild homology as the pullback of the pushforward of the structure sheaf (or more generally a sheaf of modules), and it turns out that this is in $\mathsf{D}_{\textrm{qc}}(X)$ although there's a priori no formal reason for this. The quasi-coherated version is however more natural in the context of DAG, and I was hopeful that the two agree.

@DavidBenjaminLim The stack I'm most interested in is Olsson's stack parameterizing fine log structures. In particular, an algebraic stack locally of finite presentation over the base scheme, whose diagonal is representable, locally separated, and of finite presentation. However, it is not quasi-separated. I want to be able to consider general log structures, so I don't want to place assumptions on the morphism $X\to\mathcal{Y}.$ I'm more happy to place restrictions on $X,$ but I'd prefer it to be as general as possible.

That being said, I wouldn't be opposed to hearing about mild conditions which imply $\Delta$ is qcqs (or if I'm being silly and it nearly always is qcqs or something). However again, the stack I care about isn't the nicest (it's very large), so some more typical situations might not apply.

@DavidBenjaminLim -- thanks for the comments; I've edited the question accordingly. I don't want to use $\mathbf{L}(f_\textrm{qc})^*$, since it wouldn't make sense to apply that to $\mathbf{R}f_*\mathcal{F}$ in general, so now we're only dealing with a morphism between algebraic spaces (although I'd still be interested in potential generalizations if an answer can be obtained in this situation). I know that the two pushforwards coincide when the map is qcqs, but since the stack I care about is not very well behaved, I don't want to assume this outright.

@YiningChen I resolved this issue for myself by deciding to work with general $\infty$-categories, rather than trying to work in the dg-setting, so I never did get a definitive answer to this question. However, a new paper was posted to arXiv related to this topic, titled "k-linear Morita theory," by Matteo Doni. I haven't read it in detail, so I don't know whether it answers this question, but at a glance it might be possible that the equivalence detailed therein is symmetric monoidal.