Given that you have built a standalone APK that your client has installed on the phone: when you publish, the development server does not need to be restarted. It is not involved anymore. Just bear in mind that the client might need to restart the app twice to get the update. After the first restart it should check for and download the new code/assets. After the second restart it should load the new code/assets.
When you run
The development server is only used for you while you are developing.
If your client was running the Expo app instead of your standalone app then you could potentially run your dev server in “tunnel” mode and have your client open the app from there, but that is more error prone and might be confusing to the client. Also there are a few differences between running in the Expo app and standalone (e.g. the Updates module only works in a standalone app), and since the final app that you upload to the Play Store will be a standalone app I think it’s better for the client to test with your standalone app rather than one running within the Expo app on their phone. (I hope I’ve explained this well.)