A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$

We give a bijection $\Phi$ from rigged configurations to a tensor product of Kirillov--Reshetikhin crystals of the form $B^{r,1}$ and $B^{1,s}$ in type $D_4^{(3)}$. We show that the cocharge statistic is sent to the energy statistic for tensor products $\bigotimes_{i=1}^N B^{r_i,1}$ and $\bigotimes_{i=1}^N B^{1,s_i}$. We extend this bijection to a single $B^{r,s}$, show that it preserves statistics, and obtain the so-called Kirillov--Reshetikhin tableaux model for $B^{r,s}$. Additionally, we show that $\Phi$ commutes with the virtualization map and that $B^{1,s}$ is naturally a virtual crystal in type $D_4^{(1)}$, thus defining an affine crystal structure on rigged configurations corresponding to $B^{1,s}$.