Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ?

More precisely, is there an $\infty$-topos $BG$ such that for every localic topos $\mathcal{L}$ the category of geometric morphism from $\mathcal{L}$ to $BG$ is equivalent to the category of $G$ principal bundle over $\mathcal{L}$, where a $G$-principal bundle over $\mathcal{L}$ is a locale $\mathcal{X}$ endowed with a $G$ action and an invariant map $p: \mathcal{X} \rightarrow \mathcal{L}$

such that:

1)$p$ is an open surjection.

2) The canonical map $\mathcal{X} \times G \rightarrow \mathcal{X} \times_{\mathcal{L}} \mathcal{X}$ is an isomorphism.

I am especially interested in the cases where $G= \mathbb{U}$ (the group of complex number of module $1$) and $G=\mathbb{R}$.

Of course, if $G$ is pro-discrete, then the answer is yes: it suffice to consider the infinity topos associated to the $1$-topos of continuous $G$ set. In the general case, one should look for an infinity topos of spaces endowed with a $G$ action (up to homotopy), but my knowledge of homotopy theory is not enough to see if this trivially work/does not work or if it is a difficult question...