This prior seems a bit hard to reason about and I find it more natural to put a prior on r directly. However, it appears your answer is in A better unit vector - #30 by Seth_Axen and I think you’re right (I’m a bit confused on the negative sign).
Where @Seth_Axen writes:
bijectively map (x, r) \in \mathbb{S}^n \times R_{>0} to \mathbb{R}^{n+1} \backslash \{0\} with y = r x, and use the Jacobian correction -n \log r to get the log-density \log\pi_y(y) = \log\pi_x(y/\lVert y \rVert) + \log\pi_r(\lVert y \rVert) - n\log \lVert y \rVert.
Where y \in \mathbb{R}^{n + 1}.