Hmmm, maybe we are all thinking about this the wrong way?
IF your machine has enough Z travel. . . . . .
How about just making the item with flat sides and then put it in a pair of fixtures, the first would hold the "rough" blank made from flat pieces. The second would hold the spherical shape you cut in the first fixture. I notice you have vacuum equipment so that should be doable.
Just a thought. . . . .
"If you see a good fight, get in it." Dr. Vernon Jones