Ok, so... we know from the literature how the $n$-particle version of the center of mass Calogero with coupling $k(k+1)$ arises from the reduction of the Laplace operator on $su(n)$. Namely, we have to start with functions on $su(n)$ taking values in the irreducible representation $V_{kn\lambda_1}$. This representation has highest weight $kn\lambda_1$, where $\lambda_1$ is the first fundamental weight. Similarly to the $n=2$ case, this is a symmetric tensor product of the defining $n$-dimensional representation, $\mathbb{C}_n=V_{\lambda_1}$ . Namely, one has $V_{kn\lambda_1}=\mathcal{S}(V_{\lambda_1}^{kn})$; which is the symmetric part of the $nk$-fold tensor product of $\mathbb{C}n$ with itself. The crucial and exceptional property of this IRREP of $su(n)$ is that its weight zero subspace is 1-dimensional. This is why we get a scalar differential operator (alias Calogero) from the Laplacian acting on $V_{kn\lambda_1}$-valued functions. The reduced wave function must lie in the weight zero subspace, and we obtain a matrix differential operator unless this subspace is 1-dimensional.
Now, the Dirac operator must be given as the starting point, and it must act on functions that lie in $C^{\infty}(su(n),V_{\text{spin}}\otimes U)$ where $V_{\text{spin}}$ denotes the representation space of the Clifford algebra built on $su(n)$, where the $\gamma$-matrices act, and $U$ is some other representation space of $su(n)$. What we want may work only if the following can be done. The representation space $U$ must be such that the decomposition of $V_{\text{spin}}\otimes U$ into $su(n)$ IRREPS contains the representation $V_{kn\lambda_1}$ for some $k$.
Do you see how this could happen? To take $U=V_{\text{spin}}$ works for su(2), but I do not see why this should be a good choice for any other $n$.