Your painter will need to use a paint and varnish stripper to remove the polyurethane and hopefully most of the stain. Since the texture is rough sanding isn't possible but it might be possible to carefully use a brass wire brush or steel wool while working with the stripper to help remove embedded stain and finish. This won't be easy but it is doable.
After the applied stain and finish is removed your painter can try to match the existing untouched wood. Note: It might not be possible to get the wainscoting back to its original color. Stripping will expose new wood that might not match the other wood accessories in your home.