You can technically distribute using your own license to the client, however this is not strictly the way it was designed to work by Apple.
Best practice is that they should sign up for the Enterprise program and send you the certificate and provisioning profile (or give you access to their Apple Developer portal).
Then they are responsible for distribution and licensing costs.
If you made them part of your Enterprise Program that would only really work if you only plan to work with the one client.
Alternatively you could send them the binaries and they can re sign them on their side.
this is a guide to how to re sign if required: