Then I realized that horizontally sliding, split shutters [Palomar style] would still suit the angled geometry of my faceted [tetrahedral] dome. It would just require that the shutters themselves follows the angles of the tiers of the facets just like the rest of the dome. No doubt the shutter facets would each need to be slightly deeper than the dome because they were being overlaid.
Here's yet another mock-up with the slit frame resting on top of the clamped dome segments.The main problem is that a beyond-the-zenith, cut-out would require shortened ribs. With the result that the [clamped] dome would no longer be self-supporting without the slit frame already fixed in place. Another of those chicken and egg problems.
Then I imagined a square frame carefully supported and centered at finished dome height. The ribs could then be carefully marked and cut away to match this frame. Allowing the slit frame to take its rightful place once the segments are all accurately prepared.
Rain should run down these full length shutters just as well as the rest of the dome. Without intervening rain-collecting ledges. As might well occur with small doors. If done properly the closed shutters should offer the least offense to the eye as a discontinuity of the overall form of the dome. The overlap between the two shutters would obviously need attention to stop rain from driving between them. The shutters would slide horizontally on stainless steel pipe firmly fixed at the skirt and the over-the-zenith frame.