From the file "Not my strongest suit". Maybe Steve or Jay will have better knowledge on this.
I'm not certain what solvent would have been used as the vehicle for a clay based pad (In fact I've never heard of a clay based pad) Try using mineral spirits in an inconspicous place then move to denaured alcohol. If you get into some of the stronger ones, you can get your self exposed to some nasty chemicals, and possibly stain the wood with residue.
My question is why you think that sanding the surface down and refinishing would cause more damage. If your floors are solid wood, it may be your best option.