= Solution
Since $dt=dv-dr/f$, the inverse-$F$ radial terms cancel. The <Ingoing Eddington-Finkelstein coordinates> give
$$
\boxed{ds^2=-\frac{F}{\sqrt H}\,dv^2+2\,dv\,dr+r^2\sqrt H\,d\Omega^2.}
$$
At $r=2m$, $H=1+\sinh^2\alpha=\cosh^2\alpha>0$. Thus all coefficients are analytic there, and the determinant of the $(v,r)$ metric block is $-1$, not zero. The angular area is also nonzero. This is an <analytic extension of a spacetime> through the apparent <coordinate singularity>, into the regular region $0<r<2m$.
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