Unfortunately there isn’t a way to get a single link for a playlist shared on the client review site. The client review site is intended for only the specified external users. Each client receives their own unique url to access a playlist shared to the client review site, so they will need to go by the email link that is provided directly to them.
As an internal user, from the client review site you can re-share the playlist link with a client if they lose track of their emails, but they will still receive the link by auto-generated email and not as a link provided to you that you can then put in an email.
A single playlist link is only available to internal users for the internal playlist. The clients would need to be users on your ShotGrid site.