One trick is to avoid gradients, stick to stylized flats. Then you have more colors over for AA, perhaps.
It's tricky though, since this palette has no skin tone, I it had to go in favor of bringing in another color. The palette isn't really suitable for larger motifs. Now you have to use light and shadow to give the appearance of an in-between color (the eye will calculate the local color of the subject by estimating the lighting situation).