Although I'm not aware of an existing case study on this topic, however, during my experience of contract negotiation, I have come across such situations and happy to share some tips that I usually apply. Typically, large MNCs or Government institutions like to push their contract templates and are averse to any changes.
During the beginning of the negotiation lay out on what I call the 'ground rules'. Meaning the aim of the negotiation is for both parties to explore solutions to the issues and not have an ego-tussle. The words "No" or "I don't agree" or "It cannot be done" will be used only after exploring all the available legal options. There is always something to bargain, never give away a demand for free, always ask for something in return. Try beginning with smaller issues first. In many common law jurisdictions, entering into a contract by using undue influence make the contract voidable.
This 'fee or free' battle is very common and is partly linked to organisational culture, partly to the quality of original requirements / scope of work. It isn't clear from your message whether the issues are created through customer expectations that go beyond your own, or whether there are disagreements over formal change requests. As a first step, you need to thoroughly review the scope of work and the change control procedures to determine your rights and ensure the customer is complying with relevant provisions.