Take $A = \mathbb{Z}[X^{2}]_{(2,X^{2})}[T], \ B = \mathbb{Z}[X]_{(2,X)}[T],$ where by $\mathbb{Z}[X]_{(2,X)}$ we mean localization of $\mathbb{Z}[X]$ in the prime ideal $(2,X)$.
Then $K = \mathbb{Q}(T, X^{2}), \ L = \mathbb{Q}(T,X)$, so $L/K$ is finite Galois as a quadratic extension in characteristic $\neq 2$.
It is also clear that $A,B$ are integrally closed (since ring of polynomials over integrally closed domain is integrally closed and a localization of integrally closed domain is integrally closed; alternatively: observe that $A, B$ are UFDs), and $B$ is integral over $A$.
But if we take $M = 2B + (X^{2}T-1)B,$ then $\mathfrak{m} = 2A + (X^{2}T-1)A$ (for $\mathfrak{m}\subseteq M\cap A\subsetneq A$ and $\mathfrak{m}$ is maximal since $A/\mathfrak{m}\cong \mathbb{F}_{2}(X^{2})$) and the extension $B/M | A/\mathfrak{m}$ is isomorphic to $\mathbb{F}_{2}(X) | \mathbb{F}_{2}(X^{2})$ which is purely inseparable.