Thank you both for the suggestions! @JonPridham, do you have any idea where I might access Marcy Robertson's thesis? I haven't been able to track it down.

@JonPridham Thanks for this! Are there any "standard" places where one might learn the basics about these things and their homotopy theory, or any references where they're used to circumvent issues similar to the ones I mentioned in the question?

Thanks Dmitri! I was finally able to read over your answer in detail, and this is exactly what I was looking for.

@LiamKeenan Hi Liam! I think we've talked a little about this before, but in coming back I realize I'm not sure about a few things. First, why are we actually able to commute the whole of the bar construction with $\mathsf{Mod}_{(-)}$? (In particular, I don't think we have $(\mathsf{Mod}_A)^{op}\simeq\mathsf{Mod}_{A^{op}}$, or am I wrong?) Second, what happens when we restrict to $\mathsf{Perf}_A$? I think this is where we should have an equivalence between $HH(\mathsf{Perf}_A/\mathcal{Pr}^L)$ (and/or $HH(\mathsf{Perf}_A/\mathcal{Pr}_{st}^L)$) and $THH(A)$.

@MaximeRamzi Well, $A$ for me is a commutative dg-algebra over (an ordinary ring) $k,$ so modules over $A$ are already chain complexes in some sense. I don't think I want to take chain complexes of dg-modules/chain complexes (although perhaps I'm mistaken), although I might mean take the dg-nerve of the full subcategory of fibrant-cofibrant objects in an appropriate model structure.