In the case $\mathcal{G} = (G \rightrightarrows *)$, the groupoid valued presheaf $r_1(\mathcal{G})$ is not a stack and is not $B\mathcal{G}$. You can see this because for any M, the value of the presheaf is a groupoid with only one object, while the groupoid $B\mathcal{G}(M)$ has distinct isomorphism classes of objects for each isomorphism class of $G$-bundle on $M$. In general you will always have to stackify $r_1(\mathcal{G})$ unless $\mathcal{G}$ is trivial (only identity morphisms).

Spin structures are an affine space over $H^1(S^3\setminus L; \mathbb{Z}/2)$ (mentioned in the OP). The base point mentioned in this post means they are actually canonically identified with $H^1(S^3\setminus L; \mathbb{Z}/2)$. The OP already mentioned that this latter is orientations of L, which I guess is just a form of Alexander duality.

Can you be more explicit about your spaces and maps? Spin(n) is a double cover of SO(n) and so it has a natural $\mathbb{Z}/2$-action. But what is the action on SO(d)? Maybe I am being dense, but I don't understand what you mean by $Spin \times_{Z_2} SO(d)$. Also what is the vector bundle?

@TimCampion Your advisor was clearly negligent. He should have had you read all of the Barwick-Kan papers. I think I might write him a pointed email instructing him to do better in the future ;) . (Regardless, I think you would enjoy these papers very much. Most of them are very short and to-the-point -- a rare quality in our field; Also Kan's literary style comes through strongly and is quite singular among math papers - I personally love it, though I know some hate it. Also the n-relative category model is wild! Worth knowing about!!).

Doing that for all the higher simplices too solves the first problem. The second problem is that this doesn't quite work the way we want for basically the same reason that Zhen Lin's suggestion didn't quite work. We can fix it in a way that is morally similar to what Reid Barton suggested, which is to use a slightly bigger model. So you actually need a double subdivision. This double subdivision shows up in Thomason's original work too. Some of the nuance of the Barwick-Kan work is figuring exactly which arrows to localize in the double subdivision.

(cont) If we mark all the backwards arrows as weak equivalences, then when we localize we get something where the arrows can be composed as in the original category. So this is one step closer to what we want. However there are two problems. The first is that we haven't taken care of the compositions in the original category. To handle this we need to use the 2-simplicies in the nerve and use their barycentric subdivision. This will be another poset with certain arrows marked (if you localize those arrows it is equivalent to the usual "2-simplex category" 0 --> 1 --> 2. (cont....)

@BenjaminSteinberg What these constructions do, more or less, is take a category, pass to the nerve, take the subdivision of the nerve, and then use that to glue back some simple categories together to get a poset (and these simple categories have certain arrows marked as "to-be-inverted"). For example, let's take a look at what the subdivision does to an arrow. It replaces the arrow ---> with a pair of arrows pointing towards each other ---> <--- . If we do this replacement to all the arrows in our category, then we get a new category where there are no non-trivial compositions - a poset.

Yes, that is one way to get that. If you dive into the Barwick-Kan machinery a bit more you can also see that the $\infty$-categorical localization $P[W^{-1}]$ can be taken to mean the hammock localization, whose homotopy category is the usual 1-categorical localization.

So just to clarify, the 2-Rig "Poly" is a concrete example of one which does not admit a splitting principle in the sense of the OP's question (where $K_0(R) \to K_0(R')$ is injective)?