To make the problem a bit more physically reasonable, you might consider a toy model of a hollow spherical shell of mass $M$ and radius $R$ which is spinning about its polar axis with angular velocity $\omega$ - a problem which was solved to first order in $\omega$ by Brill and Cohen in 1966.
They treat the problem as a linear perturbation about the ordinary Schwarzschild solution
$$\mathrm ds^2 = - V^2 \mathrm dt^2 + \psi^4(\mathrm dr^2 + r^2[\mathrm d\theta^2 + \sin^2(\theta) \mathrm d\phi^2])$$
which is expressed in isotropic coordinates rather than the usual Schwarzschild coordinates. We will use units in which $G=c=1$. In the interior region, $V$ and $\psi$ take the form
$$V = (1+\alpha/R) \qquad \psi =\frac{R-\alpha}{R+\alpha}$$
where $\alpha \equiv \frac{M}{2}$ is the Schwarzschild radius, and where we assume that $R>\alpha$ (otherwise we'd have a black hole). If we take the sphere to be rotating with angular velocity $\omega$, then the perturbed metric becomes
$$\mathrm ds^2 = -V^2 \mathrm dt^2 + \psi^4(\mathrm dr^2 + r^2[\mathrm d\theta^2 + \sin^2(\theta) [\mathrm d\phi - \color{red}{\Omega(r) \mathrm dt}]^2])$$
On the interior region, the Einstein equations yield that $\Omega(r)=\Omega_0$ is a constant given by
$$\Omega_0 \equiv \frac{\omega}{1+\left[3\frac{R-\alpha}{4M(1+\beta)}\right]} \qquad \beta \equiv \frac{\alpha}{2(R-\alpha)}$$
Because $\Omega(r)=\Omega_0$, the metric on the interior is related to the unperturbed metric by simple change of coordinates $\phi \mapsto \phi-\Omega t$.
The physical meaning of this is that for an observer sitting inside the shell, non-inertial effects (such as the water creeping up the sides of the bucket) are absent when the bucket is rotating at a rate $\Omega_0\neq 0$. The fact that $\Omega_0 \neq \omega$ is because the asymptotic spacetime (the $r\rightarrow \infty$ limit in the exterior region) is non-rotating, so there is a kind of "competition" between the asymptotically static spacetime and the dragging influence of the rotating shell. However, as the mass of the shell increases and $\alpha \rightarrow R$, we see that $\Omega_0 \rightarrow \omega$ and the "rotation" of the interior region becomes totally dependent on the rotation of the shell.
In other words, on the interior of a sufficiently massive spherical shell, the (locally) inertial frames are the ones rotating along with the mass. A stationary observer who sees the shell rotating will also observe non-inertial effects like the water climbing the walls of their bucket.
