After many attempts, I finally beat Big Box Two!
I found this challenge watching some Youtube video, and saw that the Big Box Two looked a lot like a design I was working on. I had to test it out! Needless to say, my initial design failed miserably. I am brand new to Cosmoteer (less than a week playing), so my first designs were not very efficient.
Not wanting to rip off your design, I did take a few pointers on efficient room placement. With a straight-up cube design, there was no way I could win. So I got creative with "detachable" parts. My cube has 2 detachable "weapon walls". This nearly doubles my firepower and acts as an excellent diversion for your Box so that it exposes its soft missile edges.
I've run the Battle Helper a few times and I always win. See below for my designs and how I progressed:
Cube (before seeing Teg's ships)
Failed to win:

Cube2 (refining design a bit to test against BigBox)
Failed to win:

Cube3 (full redesign, incomplete, went through MANY iterations, still cannot match BigBox)
Failed to win:

Cube4 (another redesign, detachable walls, 4-wide beams for better power concentration)
Winner!

I am sure I can refine it more.