I think the problem may be in the bijection. Consider instead the following bijection between (planted plane) trees with $n$ vertices and Dyck paths from $(0,0)$ to $(n-1,n-1)$: perform a depth-first search on the tree, moving $\uparrow$ in the Dyck path when you follow an edge away from the root, and $\rightarrow$ in the Dyck path when you follow an edge towards the root. Then a tree of height $h$ corresponds to a Dyck path which reaches $y=x+h-1$ but not $y=x+h$, so $A_{n,h}$ counts the Dyck paths from $(0,0)$ to $(n-1,n-1)$ which avoid the boundaries $y=x+h$ and $y=x-1$. By the referenced formula of Mohanty,

$$A_{n,h} = \sum_{k \in \mathbb{Z}} \left[\binom{2n-2}{n-1-k(h+1)} - \binom{2n-2}{n+k(h+1)}\right]$$

Then

$$\begin{eqnarray*} B_{n+1,h-1} &=& A_{n+1,n+1} - A_{n+1,h-1} \\ &=& \frac{1}{n+1} \binom{2n}{n} + \sum_{k \in \mathbb{Z}} \left[ \binom{2n}{n+1+kh} - \binom{2n}{n-kh} \right] \\ &=& \frac{1}{n+1} \binom{2n}{n} + \sum_{k \geqslant 0} \left[ \binom{2n}{n+1+kh} - \binom{2n}{n-kh} \right] + \\&& \sum_{k > 0} \left[ \binom{2n}{n+1-kh} - \binom{2n}{n+kh} \right] \\ &=& \frac{1}{n+1} \binom{2n}{n} + \binom{2n}{n+1} - \binom{2n}{n} + \\&& \sum_{k \geqslant 1} \left[ \binom{2n}{n+1+kh} - \binom{2n}{n-kh} - \binom{2n}{n+kh} + \binom{2n}{n+1-kh} \right] \\ &=& \sum_{k \geqslant 1} \left[ \binom{2n}{n+1+kh} - \binom{2n}{n-kh} - \binom{2n}{n+kh} + \binom{2n}{n+1-kh} \right] \\ &=& \sum_{k \geqslant 1} \left[ \binom{2n}{2n-(n+1+kh)} - \binom{2n}{n-kh} - \binom{2n}{2n-(n+kh)} + \binom{2n}{n+1-kh} \right] \\ &=& \sum_{k \geqslant 1} \left[ \binom{2n}{n-1-kh} - 2\binom{2n}{n-kh} + \binom{2n}{n+1-kh} \right] \\ \end{eqnarray*}$$

as desired.