The lax pullback can be written as a pseudo pullback, if the $2$-category you're working in admits path objects. We have $X\overset{\leftarrow}{\times}_Z Z\simeq X\times_Z\operatorname{Path}(Z),$ $Z\overset{\leftarrow}{\times}_Z Y\simeq\operatorname{Path}(Z)\times_Z Y,$ and $X\overset{\leftarrow}{\times}_{Z}Y\simeq\left(X\overset{\leftarrow}{\times}_Z Z\right)\times_{\operatorname{Path}(Z)}\left(Z\overset{\leftarrow}{\times}_Z Y\right).$ In this case, the category of arrows is a path object -- so I just need appropriate model structure(s) on the arrow category.

Thanks for all this information, Simon! It makes sense that the general statement is a bit too much to hope for. The particular situation I'm really interested in is more specific: $F$ is the identity, and $G$ is indeed a right Quillen functor. I'm not very familiar with Brown categories, but perhaps such things would actually apply here -- I'll need to check on this. For now, I'll leave this up in case anyone has an answer for the particular special case I'm interested in, but I'll accept this if I can manage to work something out using the nice summary you've laid out.

@HarrisonChen Thanks Harrison! I've done a deep dive into the papers you've mentioned and their references, and I think I have a much better understanding of the relationships I've been asking about. I haven't totally been able to make sense of everything (see here), but I do have a much clearer picture of the situation now. Once I figure out the remaining details to a satisfactory degree, I'll write an answer to this with what I've learned.

@QiaochuYuan Yes, you're right -- I see now that I missed the (c) assumption in the OP

Putting together results from these references, I think that if $A$ and $B$ are Morita equivalent $k$-algebras, viewed as DG-$k$ algebras in degree $0,$ they should have quasi-equivalent DG-derived categories, but not necessarily be quasi-isomorphic as DG-$k$-algebras.

I believe the key phrase here is "(derived) Morita theory." There is a wide literature on this, for example, see here, here, here, or here, to start. It may also help to reference Cohn or Cisinski and Tabuada for a quick overview of model structures on DG categories, as well as other work of Tabuada on those model structures.

@Wojowu Ah yes, that would make more sense.

Am I missing something here? If $R = k[t^2,t^3],$ then $R[t] = k[t],$ so you're just looking at the Picard group of $k[t].$