Yes, you can use any type of sugar in the bath whip. I made my sugar scrubs with the bath whip when I first started but then decided to make my own emulsified sugar scrubs with no soap in it at all.
Here is a link to a WSP recipe. I made it and really liked it. This is the one I made, but you can change the sugar, fragrance and color. If you use bath whip, it will have some chemicals added.