I will probably get push back on this, but I'm going to say it anyways.
Let me first iterated that I enjoy colaboration. Working with other people and melding ideas it's absolutely the best part of what we do. I believe in exhausting all resources, and doing everything I can to help my clients achieve ultimate success.
While I agree to most of the feedback that tells you to listen, research, show supporting data, appeal to reason, show use cases etc. In the end you, the work you do and the reputation you get doing it, will be here long after clients have moved on for another designer/agency or perhaps their business has failed.
Sometimes when it gets right down to it, if you can't do good work. You absolutely know it's a mistake, and not an arbitrary artistic disagreement. You just might need to fire that client.
In some cases this has worked out positively for agencies. For instance when I first got out of design school, an advertising agency made headlines stating They had fired Nike as their client. The CEO stated that Nike hired them do what they do best, and Nike needed to let them do that. A small time later Nike came back and agreed with the agency. In the end of the agency was very successful.
I believe in life, art advertising and perhaps even more in a quantifiable, research, and case study, fact not oppinion driven field of UX. Where fighting for the customer is more than an idea. I'ts a passion driven empathetic roll, defined by each of us, and represented by our work, results, and integrity to question what we think we know. I remind myself before every qestion regarding UX, that even with two decades of experience my opinion is my own. without solid current research, and interaction with the user I can only attest to how I feel about the product.
Scary as it can be. To have big successes sometimes you have to take big risks. Sometimes you will fail and sometimes you will succeed! The truth is, all great ideas started with a person like you and me who was willing to take a very big risk. Today those are the design heroes we look up to.
I wish you the best of luck, and ultimate success for you and your client.