Zygarde would not go on the ship since she would see it as the territory of all the legendries and taking one step on it would have all the Pokémon gang up and most certainly beat her. She would see the MC as the source of the potential apocalypses but also could prevent a other apocalypses down the road so she would test him. The chase could start outside the ship at the harbor and go though shipping containers and cranes but it starts right outside the ship that the legends that could be on the ship to watch and be like a cheer squad since the other legends that could be caught always stay on the boat and his current legends would be released to protect the boat and populated areas without needing to mention what they do.
Another idea for the hatched Pokémon could be a image but they are in a chibi like baby form that would look adorable if it can be put in when they first hatch.
plus zygarde would be similar to Rayquaza for babysitting two legendaria's from fighting and now the MC has brought over an even bigger migraine that it will not allow. The persinality of zygarde does fit to what would be needed for a legendary to come out of the wild and be in a more potentially populated location This could show how much zygarde puts importance to the environment since it is the source of her power and would hold back. By having it in a Populated area she would not hold back if the threat is great enough. After the battle the town or city could hold a festival to thank the MC with a special festival (optional).
I did have an idea that could foreshadow one of the legendary of Gen 8 and would connect this to it well "A day of darkness is coming again and you may be able to prevent the destruction it will bring" I did read zygarde was at the first day of darkness so she saw the destruction there. I do have more ideas for this but dont want to go to far if this is to big or complex to be even included or see what Brpdchainsaw thinks first if they may be interested to hear more or not.
Thank you for hearing my idea!