site under maintenance

There is maintenance mode plugin
https://www.ads-software.com/plugins/wp-maintenance-mode/

As per the plugin it
Adds a maintenance page to your blog that lets visitors know your blog is down for maintenance, or add a coming soon page for a new website.

Also User with admin rights gets full access to the blog including the front end. Your client will be able to see if logged in, while rest of the user’s cant.

yes, but there is no way to provide the customer with a personalized link that will allow him to see the site directly without logging in during the maintenance mode of the site.

You could try this plugin.
https://www.ads-software.com/plugins/hide-my-site/

thank you