Inexact labelled binary tree matching
I think that the problem is not hard, because if I understood the problem statement correctly, it can be solved in $O(|V|^2)$ time as follows:

We have two $0$-$1$-labeled rooted perfect full binary trees $(A, x)$ and $(B, y)$ with $2^k-1$ nodes. We compute the minimum number of mismatches in any isomorphism between the trees denoted by $F(A, B)$ recursively by using the $F$-values of the subtrees. We have essentially two options when choosing the isomorphism: either we swap the child subtrees of the root in one of the trees or we do not swap anything. Therefore
$F(X, Y)$ is obtained as a minimum of $F(X_{\text{left}}, Y_{\text{left}}) + F(X_{\text{right}}, Y_{\text{right}})$ and $F(X_{\text{left}}, Y_{\text{right}}) + F(X_{\text{right}}, Y_{\text{left}})$, increased by one if the $x$-label of the root of $X$ and the $y$-label of the root of $Y$ are distinct. If we use dynamic programming to optimize this recursion, we have to compute $F(X, Y)$ only once for every pair of subtrees $X$ and $Y$ of the original tree with equal heights.