After some thought, and much Hexagon-Fu, I decided to improve the bevels of my pod bay door edges. My circles didn't have much curve to them in the first place when I cut them out. So I increased the curved edges on the middle door before deleting its chamfered edge to make room for the new edge.
So that got rid of the poly artifacting. And it also gave me some practice doing the ridges, which I'll need to use big-time around the exterior airlock door when I start modeling on that brute.
The first try looked good. So I did the other two bay doors the same way.