Barry,
If I understand you correctly, then your employer doesn't change. Your current employer hires you out on a project basis to a third party to do a project. Right?
In this case I always recommend that you put down the name of your actual employer, as long as they can verify the hours & deliverables.
In case of an audit it's easier for your, because you don't have to send the audit packages to a dozen of clients, but instead you can give it to your employer for approval.
The situation is different if you are a SELF EMPLOYED PM Consultant. Then you would need to put down the name of your customers. But I don't think that you fall into this category.
So the first way it is: Put down your own company and the person you report to as contact person.