OK. If it's the case that there would be no additional; insulation between the Wagos and 'touchable metal', then you're right - that would no longer qualify as Class II ("double insulated").
As has been said, yes. In terms of electrical common sense, that would be fine (i.e.'safe'). Future eyebrows might be raised by the fact that you had earthed something which bore a Class II marking, and particularly if they knew that the manufacturer's instructions had said that the fitting "must not be earthed", but that would (at least, in my opinion) really be the problem of the owner of the eyebrows
Kind Regards, John