A trick that helped me develop my spine is, when you're feeling unsure about asserting yourself, ask yourself "Am I being unreasonable? Are they?". If you have to concede that you might be, then yeah, you should probably reconsider and try to figure out a compromise (or maybe not).
However, when you know (as in this case!) that they are being unreasonable it really helps to be your own advocate. Don't think in terms of "She's an important, difficult client and I'm worried about upsetting her." Try to steer your mindset to "What she's asking isn't reasonable. I can find a diplomatic way to fix this." That takes the situation from "should I?" to "What's the best way to?".
The first few times I really had to take a stand for myself were pretty nerve-wracking! I did it, though, and I realized that the world didn't fall apart, the other person may have been annoyed but they got over it, and once the stress passed I was not only happy with the situation but proud of myself. It gets easier each time.