It's easier than that.
The camera direction is already known, so just take the camera's frustum lines (the left and the right directions), and center each onto the AABB verts. Done.
As for the ellipsoid stuff, it doesn't seem necessary.
Edit:
A little more math, as requested
Line can be defined as: p + d * t. p is point, d is a vector, t is a scalar. Since camera angle is fixed, the direction of all vectors regarding the camera are fixed. The forward vector is fixed, the side vector is fixed, and the frustum vectors are fixed. In this way, the lines of the camera frustum have constant d terms.
To define the lines to lay upon the AABB vertices, we say that p is a vert of the AABB.
Now p and d are known, solve for t. t represents how far along in the direction of d, starting at point p, the intersection point resides.
Solving for t is done by solving for the intersection of the two lines. I'm sure you can figure this part out.