A better idea would be to use the distance between points C1 and C2 as target equation to minimize. That simplifies the numerical method. I'll try that when I have access to Mathematica next. I'm just thining out loud here.

So, my plan now is to:

1) Choose some starting point C1 on circle one.

2) Calculate the point C2 on circle two which is closest to the line through C1 and P.

3) Check if (close enough to) zero and leave the loop, or else continue.

4) Choose some other point on circle one, through a numerical method such as the secant method. Go to step 2 above.

In step 2 above, it should be possible to have a nice algebraic solution. The 3D equation of the line through P to C1 is known. And the y-coordinate of C2 is given beforehand.

So I formulate the distance between the coordinates of the line at Y=y2 (in the plane of circle 2), from the coordinates of the equation of circle 2. I derivate that expression with respect to the parametric angle alfa2. I set the derivative to zero and solve for alpha2. That gives the point on C2 which is closest to the line (in its own plane Y=y2, not orthogonally). I then calculate the distance C1 to C2 and use secant method to minimize it by choosing new points on circle 1. If the line crosses both circles, then that distance will be zero for some chosen C1.

I'm still convinced that there are better ways, so if anyone sees is it, I'd appreciate the advice!