= Solution
Stationarity makes $\partial_t\alpha=0$, while
$$
\partial_r\alpha=\frac{M}{(r+M)^2}.
$$
The left side of the <Bona--Masso slicing condition> is therefore
$$
-\beta^r\partial_r\alpha
=-\frac{M^2r}{(r+M)^4}.
$$
Its right side is
$$
-\alpha^2f(\alpha)K
=-\frac{Mr^2}{(r+M)^4}f(\alpha).
$$
Equality requires
$$
f=\frac Mr.
$$
Since $\alpha=r/(r+M)$, this is the <Stationary Schwarzschild Bona--Masso slicing function>
$$
\boxed{f(\alpha)=\frac{1-\alpha}{\alpha}.}
$$
Consequently $\lim_{\alpha\to1}f(\alpha)=0$, as expected in the asymptotically flat region $r\to\infty$.
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