calculating size of rectangle which fully obscures a sphere
How long should $i$ and $j$ be? By similar triangles, they should have length $R\times (1-\frac{R}{\lVert V\rVert})$.


I believe this is where you went wrong. If you draw the 2D version of this on a sheet of paper, it will be obvious that the edge of your triangle which touches the side of your square goes through the sphere. More so as the camera comes closer.

That line needs to be tangent to the sphere. So your large similar triangle will have its right angle at the contact point with the sphere, not at the sphere's center as you did. If my trigonometry is not too rusty, this gives

$\frac{R}{\sqrt{\lVert V\rVert^2-R^2}} \times \left(\lVert V\rVert-R\right)$