Sphere intersection occlusion (for hybrid raytracing)
Given that I didn't miss anything, you can probably cut this down to a problem in the 2D space. Viewing onto the plane defined by the center points of the spheres and your camera origin, the scene looks like this:



The spheres become circles with the center points $C_1$ and $C_2$, and the intersection circle is now only 2 points with only the closer one $P$ being interesting. The camera/eye is arbitrarily set to the point $E$.

Calculating if one point on the spheres is visible or not is easy: Simply check whether or not the angles at point $P$ between $E$ and $C_1$ respectively $E$ and $C_2$ are both greater (or equal to) 90 degree1.

If $P$ is visible, some part (e.g. at least that point) of the intersection circle is visible. Otherwise the whole intersection circle must be occluded by one of your spheres, namely the one which creates an angle of less than 90 degree.

Here is how it looks if $P$ is not visible from $E$:



You can clearly see how that point is occluded by the circle around $C_2$ and that the angle between $E$ and $C_2$ in $P$ is less than 90 degree.



1 Having an angle of exactly 90 degree means that the line between $E$ and $P$ just touches the respective circle/sphere in point $P$ as a tangent.