To close this thread off, I will try to expand Lurie's helpful comment into an answer:

First of all concerning examples of $\infty$-topoi that are locally, but not globally, of finite homotopical dimension, an easy counterexample is the slice $\infty$-topos $\mathcal{S}_{/X}$ with $X$ a space that is not finitely dominated - the fact that this must have infinite homotopy dimension follows directly from HTT 7.2.4.1, which implies that if it were finite, our space $X$ would have to be a retract of a finite CW-Complex.

I also found a different example given after [here](https://chat.stackexchange.com/transcript/message/52486803#52486803), which is very interesting as it shows that even presheaf $\infty$-categories, that are (using that the representable presheaves generate them under colimits) always of local homotopy dimension $\leq 0$, can have infinite homotopy dimension.

Now to the main part of by question, I will try to explain why this works using the definitions I gave above (in particular using only material from HTT); this is a bit more subtle than I thought at first glance, so please correct me if I am talking nonesense here. In his comment, Lurie states a fact that can be interpreted as a kind of *self-similarity* of the $p$-adic numbers: Every finite index subgroup of $\mathbb{Z}_p$ is itself isomorphic to $\mathbb{Z}_p$. This allows to show that we can transfer our knowledge that $Shv(N\mathcal{C})$ is of finite homotopy dimension onto the slice categories over representable sheaves - roughly as follows.

Remember that we had chosen the so-called *atomic* Grothendieck topology on $\mathcal{C}$. With a bit of work, we can show that the usual equivalence $\mathcal{P}(N\mathcal{C})_{/j(C)} \simeq \mathcal{P}(N\mathcal{C}_{/C})$, for $j$ the Yoneada embedding and $C \in \mathcal{C}$, restricts to $Shv(N\mathcal{C})_{/j(C)} \simeq Shv(N\mathcal{C}_{/C})$. Finally, the explicit combinatorics of our category $\mathcal{C}$ and of maps between the $\mathbb{Z}/p^n\mathbb{Z}$ allow us to identify these to be both equivalent to our topos $\mathcal{X}$, and therefore of finite homotopy dimension. Since further $\mathcal{X}$ is generated under colimits by the image of $j$ (note how colimits in the sheaf category, being a reflective subcategory of $\mathcal{P}(N\mathcal{C})$, are constructed), we see by the very definition that $\mathcal{X}$ must also be locally of finite homotopy dimension.