= Solution
Smoothness requires the polar angle $|V'(r_0)|\tau/2$ to have period exactly $2\pi$. Therefore Euclidean time must be identified by
$$
\boxed{\tau\sim\tau+\beta_0,\qquad\beta_0=\frac{4\pi}{|V'(r_0)|}=\frac{2\pi}{|\kappa_0|}.}
$$
A nontrivial integer multiple of this fundamental period still gives an excess-angle cone at the tip; it is not a smooth plane there. The static time has already been normalized at infinity because V tends to one.
For distinct charged roots, writing $d=\sqrt{M^2-Q^2}$ gives $\beta_\pm=\pi r_\pm^3/d$. These unequal periods cannot simultaneously remove both simple-root cones using one common time identification; the outer and inner static Euclidean regions are in any case separated by a band where V is negative. The physically relevant asymptotically flat exterior uses $\beta_+$. The neutral outer horizon gives $\beta_+=2\pi\sqrt{2M}$.
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