Back into Rhino to design the base. I came up with a design that allowed the piece to be viewed from all angles by using four metal structural bars to hold a "crown" which contains the projector and the computer.
For this design process I should have used a parametric modeling program like Inventor or SolidWorks. What started as a "simple" design became more complex than I anticipated, and by then I felt I was already so far deep into my design in Rhino that I just went with it. Had I used a parametric program for this step, I could have adjusted material thicknesses and scale without having to manually modify the object, which is something that happened a lot.
I've included the final object as a Rhino5 .3dm format for you to explore.