With Client Billing, the project stays in the original account it was set up in. If they’re on the CMS plan, they will only have access to the Editor (frontend content management) with a Collaborator account:
You can always transfer the project over to them and have them manage the site plan (without the need for Client Billing at all) but then you have limited ability to edit the structure of the site down the road without getting their login information or transferring the project back.
Hopefully that helps but don’t hesitate to reach out if you have any other questions!