= Solution
Work in geometrized units $G=c=1$, with the stationary <Killing vector field> $K=\partial_t$ normalized by $K^2\to-1$ at infinity. Put $q=2m\sinh^2\alpha$, $f=F/\sqrt H$, and $R^2=r^2\sqrt H$; $R$ is the <areal radius>. The <static spherical Komar mass in reciprocal radial gauge> gives
$$
M_K(r)=\frac{R^2f'}2.
$$
Indeed the future and outward unit normals to a round sphere are $n=f^{-1/2}\partial_t$ and $s=f^{1/2}\partial_r$, and $s^an^b\nabla_aK_b=-f'/2$. The <Komar mass> integral $-(4\pi)^{-1}\int s^an^b\nabla_aK_b\,dA$ is therefore $R^2f'/2$. Since $F'=2m/r^2$ and $H'=-q/r^2$,
$$
R^2f'=2m+\frac{qF}{2H},\qquad
M_K(r)=m+\frac{qF}{4H}.
$$
Taking the asymptotic limit yields
$$
\boxed{M=m\left(1+\frac12\sinh^2\alpha\right)=\frac m2(1+\cosh^2\alpha).}
$$
This agrees with $g_{tt}=-1+(2m+q/2)/r+O(r^{-2})$. The finite-radius <Komar integral> need not equal its asymptotic value, because this solution is not vacuum. Restoring $G$ divides the mass expression by $G$ when $m$ has dimensions of length.
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