I don't know if the vet would do it or not, I have never had it done. Chey-aut would be the person to talk to about that and I would assume that cost would vary with the area. Seeing that she appears to have dark eyes and black skin, I would assume that she is probably a palomino, I think that perlinos have lighter skin (not sure though). The thing that is really throwing me is the fact that it looks like the tips of her ears are darker than the rest of her and her mane and tail seem to have a reddish tinge, though that could just be dirt. That's what makes me think perlino, but I can't be sure.
The reddish tint in her mane and tail are natural. They arn't dirty spots (i promise!) she also had a couple of reddish spots here in and there like on right on top of her withers. And also the inner parts of her tail are a deep grey color. And the underside of her mane is also that deep grey color. I'll have to get some pictures of it.
I agree with the sabino but was going to mention maybe a sort of rose grey or roan? I'm not good at colors though so it's very much a guess, although I don't see perlino in her. Maybe a skunktail palomino (grey in the tail/mane) but who knows til you get her tested! Let us know if you do go that route, very interested now!