I'm also pretty sure that unlike the multiplayer campaign, in single-player, the player always goes first. The player's moves are all done immediately, and then the AI moves. I actually wish that they would change it to full round-robin, because it reduces the number of surprises. Invasions happen on the player's schedule. The AI can never interrupt part of a players planned two-country invasion with a pre-emptive invasion of their own.
Yeah it's player first, I think the option for MP style rounds should be added to the custom campaign menu. Leave the story one alone.