OK, short term methylene blue isn't effective against any diseases really, but it does help keep the fish calm (dark) during a dip and it can act as a substitute oxygen donor.
Very worrisome that the breathing seems worse after the dip, I always like to see at least some temporary relief from those, it means we know something to use to buy more time.
I remember working at a wholesale place back in the 1970s and the yellow tangs would all come in and develop erythema, similar to this. I just always assumed it was ammonia burns....thus me going down that rabbit hole.
Sorry I don't have a definitive direction for you to go with this: If it were flukes the FWD should have brought it some relief. It could be a bacterial disease, if that was systemic, it would account for the rapid breathing and red skin. Copper treatments need to run right at the point of causing toxic reactions with the fish - and those symptoms vary, but can include rapid breathing and skin issues. Finally, in some cases, the copper just doesn't control a problem fully and a protozoan just sort of festers, despite the copper being at seemingly good levels.
If it were my fish, I would keep it in copper and dose the tank with a broad spectrum, gram negative antibiotic like Neoplex or Kanaplex.
Jay