There must be some error in my formulas for the stereographic projection of the sphere and the inverse projection. However, I can't find the error. Here's what I have:
Let $S_K^n$ be the sphere with sectional curvature $K$, then the stereographic projection of $S_K^n$ to th $n$-dimensional hyperplane is:
$$ x=(x_1,...,x_{n+1}) \mapsto \frac{1}{1-\sqrt{K}x_{n+1}}(x_1,...,x_n,0) $$
and the inverse of the projection is: $$ x=(x_1,...,x_n,0)\mapsto \frac{2}{K||x||_2^2+1}\left(x_1,...,x_n,\frac{K||x||_2^2-1}{2\sqrt{K}}\right) $$
What I want to do is the stereographic projection from the north pole, where the sphere is centered at 0 with radius $R=\frac{1}{\sqrt{K}}$.
Your formulas are correct (as shown in Riemannian Manifolds: An Introduction to Curvature by Lee 1997, Formula 3.9).
Remember that to prove that $\sigma^{-1}(\sigma(x))=x$ for $x \in S^n_R$ you need the fact that $R^2 = ||x_{1:n}||^2 + x_{n+1}^2$, where $R=1/\sqrt{K}$ in your formulas.