= Solution
Writing the upper cone as $(r\cos\theta,r\sin\theta,r)$ gives metric
$$
ds_C^2=2\,dr^2+r^2d\theta^2.
$$
On any polar branch, the map
$$
\pi(s,\varphi)=\left(\frac{s}{\sqrt2}\cos(\sqrt2\varphi),
\frac{s}{\sqrt2}\sin(\sqrt2\varphi),\frac{s}{\sqrt2}\right)
$$
pulls this back to $ds^2+s^2d\varphi^2$, so it is a local isometry. Strictly, because $\sqrt2\varphi$ is not $2\pi$-periodic, this formula is globally single-valued on the universal polar cover and locally on $\mathbb R^2\setminus\{0\}$; that is the <developing map> used here.
An isometry fixing $p$ lifts locally to a Euclidean isometry fixing a lift of $p$. Compatibility with the cone's angular identification leaves only the identity and reflection across a radial line. Descending to the cone, these are the identity and reflection in the plane through the cone axis and $p$.
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