I'm sure this is very well known but I haven't found any references for this searching the internet so hence the question:

What's the neat abstract framework for obstruction theory for non-abelian Gerbes in an $\infty$-topos?

This is somewhat imprecise so I'll try to come up with a precise question so that it will be clear what's missing for me. I'll work with $\infty$-gruopoids all the while trying to make arguments which are easily adapted to the general case.

Let $X$ be a (connected) space and let $E \to X$ be a map whose fibers are connected and 1-truncated, (i.e. BG for some group $G$). This fibration is classified by a map $X \to BAut(BG)$.

Over $BAut(BG)$ we have the universal gerbe

$$BG \to BAut_*(BG) \to BAut(BG)$$

Which pulls back to

$$BG \to E \to X$$

However $BAut(BG)$ is itself a $K(Z(G),2)$-gerbe over its 1-truncation.

$$K(Z(G),2) \to BAut(BG) \to BOut(BG)$$

By analyzing the maps a bit we find out that $BAut_*(BG) \cong BAut(G)$ is the natural $BInn(G)$ gerbe over $BOut(G)$. So we have a natural map of gerbes $BAut(G) \to BAut(BG)$ over $BOut(G)$ which is in fact, up to an $Out(G)$-twist, the natural map corresponding to the canonical central extension

$$1 \to Z(G) \to G \to Inn(G) \to 1$$

Lets fix a "band" $\mathcal{B}: X \to BOut(G)$ and pullback $BAut(G) \to BAut(BG)$ of gerbes to get a map of gerbes $\mathcal{I} \to \mathcal{Z}$ over $X$ where $\mathcal{I}$ is an $BInn(G)$-gerbe and $\mathcal{Z}$ is a $K(Z(G),2)$-gerbe. We now have the following interpretations:

1. Sections of $\mathcal{Z}$ classify $BG$-gerbes banded by $\mathcal{B}$
2. Sections of $\mathcal{I}$ classify $BG$-gerbes banded by $\mathcal{B}$ equipped with a a global section.
3. The natural map $\mathcal{I} \to \mathcal{Z}$ induces on a sections map which takes a gerbe equipped with a section and forgets the section.

Now imagine for now on that the map $BAut_*(BG) \to BAut(BG)$ was the homotopy fiber (relatively over $BOut(G)$) of some universal map $BAut(BG) \to K$ for some universal fibration $K \to BOut(BG)$ equipped with a section $s: BOut(G) \to K$ (meaning that $BAut_*(BG) \cong BOut(G) \times_{K} BAut(BG)$).

If that was the case we would have a universal fiber sequence over $BOut(BG)$:

$$BAut_*(BG) \to BAut(BG) \to K$$

Lets pull this back to $X$ via the band $\mathcal{B}$ to get a (relative) fiber sequence:

$$\mathcal{I} \to \mathcal{Z} \to \mathcal{K}$$

Taking homotopy groups (or "homotopy sections") we get a long exact sequence:

$$\dots \to \pi_1(\mathcal{K}) \to \pi_0(\mathcal{I}) \to \pi_0(\mathcal{Z}) \to \pi_0(\mathcal{K}) \to *$$

Filling in the interpretations from above we might expect the following sequence exists:

$$ \dots \to \{ \text{???} \} \to \{ \text{sections of gerbes banded by $\mathcal{B}$ } \} \to \{ \text{gerbes banded by $\mathcal{B}$} \} \to \{ \text{obstructions for sections of gerbes banded by $\mathcal{B}$} \} \to *$$

Questions:

1. Is there in fact such universal $\mathcal{K}$?
2. Given (1) is true and the analysis above is correct, how does one calculate $\pi_0(\mathcal{I})$ given that $\mathcal{I}$ is itself a gerbe? There seems to be a circular trap here...

EDIT: Thanks to @YonatanHarpaz's comment it is clear that $\mathcal{K}$ can not exist as stated. I hope that this does not destroy the hope for a categorical framework for such obstructions.

Could there be a way to generalize and instead of requiring $\mathcal{K}$ to be a "sheaf" (meaning an object in the Topos) we could require it to be something else? Perhaps a certain "higher-category" parametrized by the Topos?

So the question is now:

Main question: Is there any hope for a categorical formalism for higher non-abelian group cohomology which would include an obstruction theory for non-abelian gerbes?