For the future, it usually helps when dealing with difficult people to have an alternative plan up your sleeve to present to them if you really can't go along with what they want, and to practice offering the alternative to make it sound as if it is their idea and would be the most flattering style or look they could ever have thought of.
For instance, you could say you can see that they are aiming for a certain look (might help to include the name of some actress about 10 - 15 years younger than the customer, whom the customer vaguely resembles) and you think they are so right, that look would be marvelous on them, then describe what would really be suitable for them, tell them it would bring out their spectacular eyes/lovely jawline/fantastic eyebrows, shape of face, blah blah. Then proceed to tell them that the best way to achieve this is your way.
Or try saying (through gritted teeth) that you would love to keep them as a client and that you really couldn't take their money for something that in your experienced opinion would end up making them unhappy (massaging their feelings instead of focusing on the condition of their hair). Some difficult people like to think others are taking care of their feelings for them.
Alternatively, you need to adopt an air of authority (not bossiness but knowledgability) and tell them with a regretful tone of voice that a chemical process at that time would damage their hair -- try not to say 'damage it even more' because some people are very vain and touchy.
The worst thing you can do is take the abuse you are receiving personally and to focus on that, get your back up, and show them that you are getting upset. Some customers get out of bed on the wrong side and take out their frustrations on people they are buying services from. Try to take a deep breath, force a little smile, and speak in a cool and calm tone of voice.