$\newcommand{\al}{\alpha}\newcommand\la\lambda\newcommand\R{\mathbb R}$Such a construction of $M$ and $\al$ is always possible. 

Indeed, take any complex $\la$. Interchanging columns and rows of the matrix $A-\la I_{3n}$, we see that $\la$ is an eigenvalue of $A$ iff 
\begin{equation*}
	D(\la):=\begin{vmatrix}
	-\la I&I&0\\
	0&(1-\la)I&-\al I \\ 
	0&B&C-\la I
	\end{vmatrix}=0,
\end{equation*}
where $|\cdot|$ denotes the determinant, 
\begin{equation*}
	B:=(M+H)^{-1},\quad C:=B(M-2\al I),
\end{equation*}
$I:=I_n$. 

Note that $D(\la)$ is the determinant of block-triangular matrix, so that 
\begin{equation*}
	D(\la)=(-\la)^n
	\begin{vmatrix}
	(1-\la)I&-\al I \\ 
	B&C-\la I
	\end{vmatrix}. 
\end{equation*}
So, $D(1)\ne0$, since $B=(M+H)^{-1}$ is nonsingular. 

So, without loss of generality (wlog), $\la\ne1$, and then, by ["The general case"][1], 
\begin{equation*}
\begin{aligned}
	D(\la)&=(-\la)^n(1-\la)^n\,|C-\la I-B((1-\la)I)^{-1}(-\al I)| \\ 
&	=(-\la)^n(1-\la)^n\,|B(M-2\al I)-\la I+\al(1-\la)^{-1}B| \\ 
&	=(-\la)^n(1-\la)^n\,|B|\,|(M-2\al I)-\la(M+H)+\al(1-\la)^{-1}I| \\ 
&	=(-\la)^n(1-\la)^n\,|B|\,d(\la), 
\end{aligned}
\end{equation*}
where 
\begin{equation*}
	d(\la):=|(1-\la)M-\la H+\al((1-\la)^{-1}-2)I|. 
\end{equation*}

So, $\la$ is a nonzero eigenvalue of $A$ iff $d(\la)=1$. 

By diagonalization, wlog the matrix $H$ is diagonal, with (say) real $h_1,\dots,h_n$ on its diagonal. Letting now $M$ to be diagonal as well, with positive real $m_1,\dots,m_n$ on its diagonal, we see that 
\begin{equation*}
	d(\la)=\prod_{i=1}^n f_{\al,h_i}(\la,m_i),
\end{equation*}
where $f_{\al,h}(\la,m):=(1-\la)m-\la h+\al((1-\la)^{-1}-2)$. 

For $\la\ne1$, the equation $f_{\al,h}(\la,m)=0$ for $\la$ is equivalent to a quadratic equation, with certain roots $\la_1=\la_1(\al,h,m)$ and $\la_2=\la_2(\al,h,m)$, say. 
Moreover, taking any $\al\in(\max(0,-h),\infty)$ and then choosing $m=\frac{h^2+4\al^2}{4\al}$, we get 
$\la_1=\la_2=\frac h{2\al+h}\in(-1,1)$. 

So, for any real $\al>\max(0,-h_1,\dots,-h_n)$ we can find positive real $m_1,\dots,m_n$ such that all the roots $\la$ of the equation $d(\la)=0$ are in the interval $(-1,1)$. 

Thus, we will have all the eigenvalues of $A$ in the interval $(-1,1)$. $\quad\Box$ 

  [1]: https://www.statlect.com/matrix-algebra/determinant-of-block-matrix
  [2]: https://i.sstatic.net/C71Rq.png