"This $1/r^2$ relationship is tied to there being three spatial dimensions."
This is an assumption. The inverse square is based on observation. If there are 10D, then the inverse square (IS) is based on 10D, not 3D. Note that the IS is 2D. It's not like Newton said, "Hmmm, there are three dimensions, so I should make my inverse 2D." Further, if we change r2 to r8, then we also should change E=mc2 to E=mc8 and every other formula in physics that has a vector squared.
For all we know, r2 = x2 + y2 + z2 + (extra squared dimensions).