You can dye the hair while leaving it rooted. I wouldn't put the whole pony in though. Staining may not be a problem because she will be painted black, but the dye will seep through the paint. So if you use light colored paint on her anywhere, then the dye can seep through the black paint and into the light paint.
I would just dunk the mane and tail in. I might even go so far as to smear the head and rump with Vaseline in order to protect it from the dye.