For such saddle pads, I first brush most of the hair off with a damp, rough brush (the same I use for brushing my horse's body), so that it doesn't get stuck in the washing machine. And then I just wash it in the washer, using no deteregents or a mild, anti-allergic deteregent without scenting. It should be completely ok - that's how I've always washed my saddle pads and none of them have been damaged that way.
By the way, you should avoid using a saddle pad until it has lots of sweat and hair on it. It can cause rubs and is generally unpleasant for the horse, especially, if he has a sensitive skin.
I have come a long way, to surrender my shadow to the shadow of my horse.