Mihai has pointed out that his proof includes a partial version of a Poncolet's Porism for ellipsoids inside tetrahedron inside ellipsoids, it might be interesting to see how far that can go. My original question could also be generalised to simplices in higher dimensions. But I think my original problem is solved (the best way to reduce the none-tight case is perhaps to shrink the base of the tetrahedron to make it tight, then shrink the cross-section of the Bloch sphere to a tight ellipse round that.) Thanks for all the fantastic suggestions, and in particular for the name of the Porism.

Thanks. I agree that that ellipse looks like it won't fit in a triangle, but I don't agree that it fits inside the tetrahedron. If you try to make a thin tetrahedron like that then a typical cross-section through it will be a diamond which your ellipse wouldn't fit into. If you have Mathematica, the code at pastebin.com/UCAYTPpc shows the sort of tetrahedron I think your talking about and lets you look at cross-sections through it.