A bijective proof of an identity of Berkovich and Uncu

The BG-rank BG($\pi$) of an integer partition $\pi$ is defined as $$\text{BG}(\pi) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $\pi$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+\nu}(k,q)=q^{2k^2-k}\left[\begin{matrix}2N+\nu\\N+k\end{matrix}\right]_{q^2}$$ for any integer $k$ and non-negative integer $N$ where $\nu\in \{0,1\}$, $B_N(k,q)$ is the generating function for partitions into distinct parts less than or equal to $N$ with BG-rank equal to $k$ and $\left[\begin{matrix}a+b\\b\end{matrix}\right]_q$ is a Gaussian binomial coefficient. In this paper, we provide a bijective proof of Berkovich and Uncu's identity along the lines of Vandervelde and Fu and Tang's idea.
Comment: 18 pages, 8 figures. To appear in S\'eminaire Lotharingien de Combinatoire