Devra Garfinkle developed such a correspondence to pairs of domino tableaux in a series of Compositio Mathematica papers in the early 1990s. (The first & third, but curiously not the second, are available through EUDML: https://eudml.org/doc/90031 and https://eudml.org/doc/90244.) More succinct summaries are given in the work of McGovern, van Leeuwen, Shimizono, Pietraho, Taskin, etc.
The $W(B_2)$ example you request doesn't involve any of the horizontal/vertical domino rotations that make this theory tricky, just a little bumping. Here are the 8 domino tableaux pairs in the order you gave the $W(B_2)$ elements. Note that all but the 5th and 6th are involutions, so their left and right tableaux are the same in all but those. [If someone knows how to TeX these into nice domino pictures, please do so and let me know how you do it.]
\begin{gather*} \left(\begin{matrix} 0 & 1 & 1 & 2 & 2 \end{matrix}, \quad \begin{matrix} 0 & 1 & 1 & 2 & 2 \end{matrix}\right), \quad \left(\begin{matrix} 0 & 1 & 1 \\ 2 & 2 \end{matrix}, \quad \begin{matrix} 0 & 1 & 1 \\ 2 & 2 \end{matrix}\right), \\ \\ \left(\begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}, \quad \begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}\right), \quad \left(\begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}, \quad \begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}\right), \\ \\ \left(\begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}, \quad \begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}\right), \quad \left(\begin{matrix} 0 & 2 & 2 \\ 1 \\ 1 \end{matrix}, \quad \begin{matrix} 0 & 1 & 1 \\ 2 \\ 2 \end{matrix}\right), \\ \\ \left(\begin{matrix} 0 & 2 \\ 1 & 2 \\ 1 \end{matrix}, \quad \begin{matrix} 0 & 2 \\ 1 & 2 \\ 1 \end{matrix}\right), \quad \left(\begin{matrix} 0 \\ 1 \\ 1 \\ 2 \\ 2 \end{matrix}, \quad \begin{matrix} 0 \\ 1 \\ 1 \\ 2 \\ 2 \end{matrix}\right). \end{gather*}
(The domino tableaux for $W(C_2)$ also have dominoes labeled 1 and 2, but no 0 square.)