True colors often pop out during wedding planning time, and I think it's perfectly reasonable for OP to judge her sister. I sure am. I may be in the minority here but I think the amount that you're being asked to spend on this wedding is a lot (amounts are relative so the point is, it's a lot for the OP and that should be good enough for all of us). I also think brides who dictate hair styles and make up are going too far in their requests/mandates. I think your sister has already crossed the line of good behavior. After this wedding, I wouldn't be surprised if you distanced yourself from her. I would, after all this hoopla over your hair, as well as that tacky shower/tea she ordered you to be a part of.
However, even though I personally have brightly dyed hair, I think you should get the cut, and hold off on the color. Why? Because your sister might be the type of person who throws in your face what you did and didn't do for her when it came to her "special daaaay" and do you really want this thrown in your face? When it comes down to it, if you can take the high road, then you'll know that you did the right thing and no one can come down on you for it. And in the end, if she still comes down on you for merely cutting your hair, if it still bothers her for years on out, then you'll know for sure that your values don't align. And then maybe that'll be a reason to distance yourself.