You can do exactly as you describe. When you move the site from your server to the client’s you’ll need to follow a few simple instructions, but the process is typically very smooth.
EE’s license is per installation, not per domain. This means that EE is not “locked” into a specific domain. As long as you only have a single live/public installation per license, you’re fine. There is a “grace period” if you need to switch servers where you can have two installations up during the move. And, of course, you’re allowed to have a local/test installation in addition to the live installation at no extra cost.
Overall we’ve tried to make EE’s license policies as friendly as possible to the real world needs of developers.