Stability of the surjectivity of endomorphisms and isometries of $\mathcal{B}(H)$

We determine the largest positive number $c$ with the property that whenever $\Phi,\Psi$ are endomorphisms, respectively unital isometries of the algebra of all bounded linear operators acting on a separable Hilbert space, $\| \Phi(A)-\Psi (A)\|<c\| A\|$ holds for every nonzero $A$ and $\Phi$ is surjective, then so is $\Psi$. It turns out that in the first case we have $c=1$, while in the second one $c=2$.