How to remove gloss paint incorrectly applied to wainscoting over gloss paint
Rip out and replace the wainscoting.  Wainscoting generally comes pre-primed.  For a little more at a better lumber yard, you can get a true plywood panel.  The big box stores will have MDF, which is much less durable.

You might be able to compel the contractor to do this.... but given the story above, you probably don't want them doing it.