Okay, there's another bug that fixes this bug for some reason.
If I upgrade my commander to a privateer, it scrambles the mercenaries and other ship designs. Hiring one mercenary gets you a different one, and the ship designs in the shipyard are in the wrong groupings, like a small survey ship as part of a tiny kinetic group. However, this lets it get to the next turn.
I'm not happy though because I've given up on using the privateer so I can avoid this bug but now it seems like it's the only way to progress this save. Or I could just quit, but I don't know if this will just come up again in another game.