I'm looking for examples of Riemannian manifolds M of dimension $\geq 2$ such that the isometry group $Isom(M)$ contains as subgroup a finite Coxeter group $G$ such that $Tor(Z(G))$, the torsion group of the center of $G$, contains an element of order $\geq 3$.
Such a thing cannot exist, as the center of a Coxeter group $(W,S)$ is always an elementary abelian 2-groups.
Indeed, it is easy to reduce to the irreducible case, that is: If $S=S_1\cup \ldots S_k$ is a decomposition of $S$ into irreducible components, then $W= \langle S \rangle \cong \langle S_1 \rangle \times \cdots \times \langle S_k \rangle$, hence $Z(W) = Z(\langle S \rangle) \cong Z(\langle S_1 \rangle) \times \cdots \times Z(\langle S_k \rangle)$.
Now, for an irreducible Coxeter group $(W,S)$, if $W$ is infinite then its center is always trivial, and if $W$ is finite, it can be trivial or of order 2 (see e.g. exercise 1 in Section 6.3 of "Reflection Groups and Coxeter Groups" by James E. Humphreys. (There are perhaps better, and certainly older references for this, but I'll leave it to others to dig those up).