When $k=2$ and $n$ is even,
$$
D_{k,r} S^n \simeq BB_{r+} \wedge S^{rn}
$$
is the $rn$-fold unreduced suspension of the classifying space of the braid group $B_r$ on $r$ strings. I once cited Cohen-Mahowald-Milgram ``The stable decomposition for the double loop space of a sphere'' (1978) for this, and if I recall correctly they credited Arnold for the idea. Hence $rn$ is the maximal number of suspensions that can appear in this case.

Edit: For $k=2$ and $n=2$ the map $F(\mathbb{R}^2,r) \times_{\Sigma_r} \mathbb{R}^{2r} \to \mathbb{R}^{2r}$ taking $[z,\xi] = [z_1, \dots, z_r, \xi_1, \dots, \xi_r]$ to
$$
(\sum_i \xi_i, \sum_i z_i \xi_i, \dots, \sum_i z_i^{r-1} \xi_i)
$$
trivializes the source as an $\mathbb{R}^{2r}$-bundle over $F(\mathbb{R}^2,r)/\Sigma_r$. Here the $z_i$ and $\xi_i$ in $\mathbb{R}^2$ are viewed as complex numbers, and the products in the displayed formula are formed in $\mathbb{C}$. A similar formula works for any even $n$, by taking the direct sum of $n/2$ copies of this $\mathbb{R}^{2r}$-bundle. Hence the Thom complex of this bundle, which is your $D_{2,r} S^n$, is the $rn$-th suspension of $(F(\mathbb{R}^2,r)/\Sigma_r)_+ \simeq BB_{r+}$. I recalled this in Proposition 9.10 of my paper "Topological logarithmic structures" GTM 16 (2009).

For $k=2$ and $n=1$, or more generally for $n$ odd, the stable splitting of $\Omega^2 S^3$ is closely connected to the story of Brown-Gitler spectra.

Regarding the case when $k$ is a higher power of $2$, you might first try $k=4$ and identify $\mathbb{R}^4$ with the quaternions. However, in that case the quaternionic Vandermonde matrix might well be singular. For instance, with $r=3$ and $z = (i,j,k)$ the matrix
$\begin{pmatrix} 1 & 1 & 1 \\ i & j & k \\ -1 & -1 & -1 \end{pmatrix}$
annihilates $\xi = (1,0,1)$, so Arnold's idea does not extend to this case.