Thanks for all your help Skorche! I do have one last question though (hopefully)...

Can I decompose a nonconvex mesh into a series of smaller convex object, then use each existing penetration depth vector as the collision normals? It seems like that would converge to the correct solution over a few frames, which should be fine in practice.

Never mind, that seems to be the correct method. Even if I were to get a nonconvex penetration depth solution working, I found plenty of cases where that isn't the best solution. The only workaround is finding a method that looks at both the minimum penetration depth AND the minimum object rotation, and there's no way I'm going through all that trouble when there are faster and easier solutions.

Using a bunch of connected convex objects seems like it will converge on the correct solution in all the cases I've looked at.