Finding a permutation matrix for almost identical permuted matrices

Question

Given $A= \begin{pmatrix} 4 & 2 & 3 & 5\\ 3 & 3 & 1 & 0\\ \end{pmatrix} $ and $B= \begin{pmatrix} 3 + \epsilon_1 & 3 & 1 & 0\\ 4 & 2+ \epsilon_2 & 3 & 5 \end{pmatrix} $
How can I compute the permutation matrix?
I.e. find a Doubly stochastic matrix that satisfies
$A^T *P\approx B^T$
e.g. here P is $P= \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} $
edit: I'm looking for a closed form solution, and not iteratively finding the "most similar row" in each corresponding matrix