Apple only allows an app ID to be created by one developer account. Because your client already has that application ID, you cannot create another one. If Apple allowed this, you would have app ID conflicts with multiple developers trying to submit apps that have the same app ID.
What you should do is create your developer certificate under the client's organization account, then have your client add your certificate to their provisioning profile. Then you can download and use this provisioning profile to test the app on devices. Or, if you are on XCode 7 or later, you should be able to create your own developement provisioning profile with your devices under that team account. The key is that you need to be doing everything under the client's Team account, not your personal account.
I would not recommend your client share his credentials or make you a team agent. That's granting you too much power / control as a developer and if I were you, I wouldn't want it. I would hate to have that access if someone else went in an did something like revoking their certificate, because I know they'd be suspecting me even if I didn't do it. Use the apple developer account roles the way they were intended and you should be fine.