A possible conceptual issue here is the distinction between the following (very closely related) concepts:

- A conformal map between two Riemannian manifolds is a map $\phi : (M,g) \to (N,h)$ such that $\phi^* h = e^{2 \sigma} g$ for some scalar function $\sigma$.
- Two Riemannian metrics $g_1,g_2$ on the
*same* smooth manifold $M$ are conformally related if $g_2= e^{2 \sigma} g_1$ for some scalar function $\sigma$.

The relationship between them is straightforward: $g_1,g_2$ are conformally related if the identity map $\mathrm{id}_M:(M,g_1) \to (M,g_2)$ is conformal, and a map $\phi: (M,g) \to (N,h)$ is conformal if $g, \phi^* h$ are conformally related.

Since $\mathbb S^3$ and $\mathbb R^3$ are distinct manifolds, what you really want to show is that there is a conformal *map* $\mathbb R^3 \to \mathbb S^3$, the most likely candidate for which is (as the question linked in the comments suggests) just the inverse of stereographic projection.

Alternatively, you can use the stereographic projection to identify $\mathbb S^3 \setminus \{ p \}$ with $\mathbb R^3$, so that you have two metrics on $\mathbb R^3$: the Euclidean metric $\delta$ , and the metric $g$ obtained from $\mathbb S^3$ via this identification. The problem is then in the form you originally posed it: show that $ g = e^{2 \sigma} \delta$ for some $\sigma$.

The relationship between the two concepts means that the calculation you need to do will be the same either way.