Customer sevice problems

What is the name of your state (only U.S. law)? CA.
I hope this is the appropriate forum for this question, please excuse me if not.
We are a very small business and we have a great customer care reputation, well, almost.
We have a customer that we have bent over backwards to satisfy with countless hours on the phone and nearly 100 emails. However, whenever we attempt to address the customer complaint, the complaint changes, a moving target. This has happened to us before, and the customer was from the same country (Australia).
We have recently found rumors on the internet that this current customer has copied a product made in the USA and has copies made in China, which are sold world wide.
We suspect that the same is about to happen to us, and that our customer service department is under attack to try to discredit our customer service.

What should we do, or not do in this situation. We have been trying to be as professional as possible with our customer care but we are afraid we will make a fatal mistake in dealing with this customer that will discredit our customer care service. We know we can do little about product copying since we do not have the funds to defend our product internationaly, but our customer care reputaion sells our products, we are very protective of that.

Thank you for your advice.

Don't know your product, don't know the problems.... But, I would write a single document that you would both agree to that lists the problems, detailed descriptions and a place for a resolution description and date you addressed/resolved.

Get the list from them, put it in this format, and then each time you resolve somthing fill in the inforation, date & sign and send them a copy. Nothing gets added until the list is completed. Keep it all on the same form so that everytime something oes get completed, they see all the items--completed ones and open ones. Keep them focused on completing all the listed problems not new ones that they want to add.