I sympathise. I can only tell you what I do, and it's a useful phased approach that has worked for me over the many years I have employed the concept!
Step 1 - Explain that you are designing for the end user - Remind them that the design is this way not because you want it this way, and it shouldn't change because the client (or his wife/nephew/postman) doesn't like it. The design is like this because it has been designed for the end user and that you are trained, and experienced, in understanding what that end-user wants. You can cite A/B testing, user feedback, surveys and more to attempt to convince them that you actually understand better than they do what the end user wants to see, and more importantly what will drive them to the client's goal.
Step 2 - Suggest a third alternative/compromise in a positive way - "I love you idea, why don't we take this from it, and add it to my initial concept and now it's perfect!" Sometimes you can take "essence", or "energy" or any other thing that lends itself to the alteration of a few pixels, or none at all haha. Be very careful that this doesn't end up in a forever repeating loop (see step 4!) - In fact I would consider omitting this step entirely, it rarely works out, you might be better off....
Step 3 Consider going fully uncompromising? - Gauge the client. If you can get away with it, you might consider going full "Michelangelo" - that is to say totally uncompromising. It can work. You have to commit to it fully. Full prima donna, Simply inform them you are an expert, and frankly insulted that you are being questioned at all, and threaten to walk, more usually though....
Step 4 - Repeat steps 1 and 2 three times and then totally give in. Sad but true. This rule has saved my sanity on several occasions. It's the best piece of advice I can give you. Do everything reasonably in your power to push the design you believe in past the post, but if the client pushes back three times, simply give in. Completely.
Don't attempt to compromise that will wind you up even more. Just let them design it themselves. I have even taken a laptop to the client and sat with them "moving it left", "making it greener", this has led to some frankly hilarious results. Not my problem. I tried my best, three times, and pushing back a fourth time would simply end the relationship. I can get paid, guilt free, and sometimes I even have good work I can reuse! Naturally this work doesn't make it into my portfolio!