Forget the client for a moment. If you make more than $100k in a year you'll need Pro, otherwise you can use free.
Now think about the client. What version of Unity do they want you to use? If they are happy publishing a game that has a Unity splash screen, then maybe you can use free for developing the game. This kind of thing is usually mentioned in the contract between you and the client. ("Developer will make game using Unity Pro and deliver executable to client for publishing"). Note that if the client doesn't want a Unity splash screen, then you'll need Pro and the iOS Pro and Android Pro add-ons, for $4500.
If you ship the Unity *project* to the client as the final deliverable, and they make the mobile executables then they will need the Pro tools. In which case you probably need the Pro tools too.
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