A simple brute force method worked (even though I'm not happy with this).

Let $\zeta=\xi+i\eta$ be a non-positive eigenvalue of $M$ and
$\left[\begin{matrix} x & y & z \end{matrix}\right]^T$ be
a corresponding eigenvector.
This gives equations
\begin{align}
Ax + By \qquad &= \zeta x \\
B^Tx + Cy - B^Tz &= \zeta y\\
-Ax -By + Az &= \zeta z
\end{align}
From the first and the third, one obtains
$$z = -\zeta(\zeta-A)^{-1}x
\mbox{ and } x-z = (1+\zeta(\zeta-A)^{-1})x.$$
To ease the notation, put $A(\zeta):=1+\zeta(\zeta-A)^{-1}=(2\zeta-A)(\zeta-A)^{-1}$.
From the second,
$$y = (\zeta-C)^{-1}B^T(x-z) = (\zeta-C)^{-1}B^T A(\zeta) x.$$
By combining with the first, one obtains
$$(\zeta-A)x = By = B(\zeta-C)^{-1}B^T A(\zeta)x.$$
Note that the imaginary part of $(\zeta- C)^{-1}$ is
$$\Im\frac{1}{\zeta- C}=\Im\frac{\bar{\zeta}-C}{|\zeta-C|^2}
=-\eta\frac{1}{|\zeta-C|^2}.$$
Take the inner product with $A(\zeta)x$ and look at the imaginary part:
$$\Im \langle (\zeta-A)x,A(\zeta)x\rangle
= -\eta \langle B|\zeta-C|^{-2}B^TA(\zeta)x,A(\zeta)x\rangle.$$
Now, since
\begin{align*}
A(\bar{\zeta})(\zeta-A)
&= \frac{(2\bar{\zeta}-A)(\zeta-A)^2}{|\zeta-A|^2} \\
&=\frac{2|\zeta|^2\zeta-4|\zeta|^2A+2\bar{\zeta}A^2-\zeta^2A+2\zeta A^2-A^3}{|\zeta-A|^2},
\end{align*}
one obtains
$$2\eta\langle \frac{|\zeta|^2-\xi A }{|\zeta-A|^2}x,x\rangle
=-\eta \langle B|\zeta-C|^{-2}B^TA(\zeta)x,A(\zeta)x\rangle.$$
Hence, unless $\eta=0$,
$$\xi\geq\frac{|\zeta|^2}{\|A\|}>0.$$
Let's deal with the case $\eta=0$.
Suppose for a contradiction that $\xi\le0$. Then
\begin{align*}
\langle A(\xi)(-\xi+A)x,x\rangle
&= \langle B(-\xi+C)^{-1}B^T A(\xi)x,A(\xi)x\rangle\\
&\le \langle BC^{-1}B^T A(\xi)x,A(\xi)x\rangle\\
&< \langle A A(\xi)x,A(\xi)x\rangle,
\end{align*}
but this is in contradiction with the fact that
$A(\xi)\succ0$ and $-\xi + A\succeq A(\xi)A$.

ADDED:
We assume $\left[\begin{smallmatrix}
A_0 & B_0 & \\
B_0^T & C_0\end{smallmatrix}\right]\succ 2\epsilon$ and will show
that any eigenvalue of
$\left[\begin{smallmatrix}
A_0 & B_0 & \\
B_0^T & C_0 & -B_0^T \\
-A_0 & -B_0 & A_0
\end{smallmatrix}\right]$ has its real part at least $\epsilon$.
To ease notation, we consider $A=A_0-2\epsilon$ and $C=C_0-2\epsilon$
instead of $A_0$ and $C_0$.
By the IVT trick, it suffices to show
there is no solution for
$$\left[\begin{matrix}
A+2\epsilon & B & \\
B^T & C+2\epsilon & -B^T \\
-(A+2\epsilon) & -B & A+2\epsilon
\end{matrix}\right]
\left[\begin{matrix}
x \\ y \\ z
\end{matrix}\right]
= (\epsilon+i\eta) \left[\begin{matrix}
x \\ y \\ z
\end{matrix}\right],$$
with $\left[\begin{smallmatrix}
A & B & \\ B^T & C\end{smallmatrix}\right]\succ 0$,
$\epsilon>0$, $\eta\in\mathbb{R}$, and
$\left[\begin{smallmatrix} x & y & z \end{smallmatrix}\right]\neq 0$.

The first row + the third row: $-\epsilon x + (A+\epsilon) z = i\eta (x+z).$
Hence
$$x-z=(1-\frac{\epsilon+i\eta}{A+\epsilon-i\eta})x
=\frac{A-2i\eta}{A+\epsilon-i\eta}x=:A(\eta)x.$$
Together with the second row:
$$y=-(C+\epsilon-i\eta)^{-1}B^T(x-z)=-(C+\epsilon-i\eta)^{-1}B^TA(\eta)x.$$
It follows that $x\neq0$.
Together with the first row:
\begin{align*}
(A+\epsilon -i\eta) x
= -By
= B(C+\epsilon-i\eta)^{-1}B^TA(\eta)x
\end{align*}
and so
\begin{equation}\tag{$\ast$} \langle (A+\epsilon -i\eta) x,A(\eta)x\rangle
= \langle B(C+\epsilon-i\eta)^{-1}B^TA(\eta)x, A(\eta)x\rangle.\end{equation}
Do some calculations:
$$A(\eta)^*(A+\epsilon -i\eta)
= |A+\epsilon-i\eta|^{-2}(A+\epsilon-i\eta)^2(A+2i\eta),$$
$$(A+\epsilon-i\eta)^2(A+2i\eta)
= A^3 + 2 \epsilon A^2 + \epsilon^2 A + 3 \eta^2 A
+ 4\epsilon \eta^2 + 2i\eta (\epsilon A + \epsilon^2 - \eta^2),$$
$$A(\eta)^*B(C+\epsilon-i\eta)^{-1}B^TA(\eta)
=A(\eta)^*B\frac{C+\epsilon +i\eta}{|C+\epsilon-i\eta|^2}B^TA(\eta),$$
$$A(\eta)^*B\frac{C+\epsilon}{|C+\epsilon-i\eta|^2}B^TA(\eta)
\preceq A(\eta)^*B C^{-1} B^T A(\eta) \prec A|A(\eta)|^2,$$
$$A|A(\eta)|^2 = |A+\epsilon-i\eta|^{-2}(A^3+4\eta^2A).$$
Look at the real part of $(\ast)$: for $w=|A+\epsilon-i\eta|^{-1}x$,
$$\langle(2\epsilon A^2+\epsilon^2A+4\epsilon\eta^2)w,w\rangle
< \langle \eta^2Aw,w\rangle.$$
Look at the imaginary part of $(\ast)$:
as $\eta\neq0$ from the previous inequality,
$$\langle(\epsilon A + \epsilon^2 - \eta^2)w,w\rangle
=\langle A(\eta)^*B\frac{1}{|C+\epsilon-i\eta|^2}B^TA(\eta)x,x\rangle/2
\geq0.$$
Combine the last two:
\begin{align*}
2\epsilon \|Aw\|^2+\epsilon^2\langle Aw,w\rangle +4\epsilon\eta^2\|w\|^2
&< \eta^2\langle Aw,w\rangle\\
&\le (\epsilon \frac{\langle Aw,w\rangle}{\|w\|^2} +\epsilon^2)\langle Aw,w\rangle\\
&\le\epsilon\|Aw\|^2 +\epsilon^2\langle Aw,w\rangle.
\end{align*}
We arrive at a contradiction.