The owner of the developer account needs to be an admin of the enterprise in order to be able to publish a self-hosted app on Google Play. Therefore you will need to ask your customer to create a developer account using their admin account and upload the self-hosted app with this account.
Then here are the steps in the Google Play Console:
create a new app,
in "Pricing & distribution > Managed Google Play" select "Turn on advanced managed Google Play features",
and then select "Privately target this app to a list of organizations"
The app should now be targeted to one organization.
go to "App releases > Production track > Manage" and create a new release,
select "I am uploading a configuration for an APK hosted outside of Google Play",
upload your JSON file.
If you want to test the app before sending it to your customer you can create a test organization using the Android Management Experience. Just make sure to use a different package name for the app since package names must be unique in Play.