Fair use does not make a work public domain; it merely helps the creator of the derivative work defend their case in court. But neither the original nor the derivative becomes public domain after a successful fair use defense.
Not a lawyer, of course, but I think slapping the Getty logo on a work claiming "fair use" and then releasing the work under public domain would be a case of misrepresentation, because Getty still has a copyright claim on your work. Regardless of the copyright status, it's still a clear trademark violation to me.
You can produce a public domain work using content that you have fair use rights to. The original owner of the content you are using fairly has no claim of ownership. You would have to assert that right in court if the owner of the copyright came after you, but that does not preclude the possibility of making a public-domain work with other copyrights used in fair use.
Obviously, that would not entitle anyone to rip those elements from your work and use them in a way that was not fair use. The Getty watermark could fall into this category: public domain pictures using the watermark fairly (for transformative commentary/satire purposes) could go into the network, which uses that information to produce infringing images.
Trademarks are a different story, but trademark protections are a lot narrower than you might think.
The point is that it's very conceivable that the neural network is being trained to infringe copyrights by training entirely with public-domain images.
Not a lawyer, of course, but I think slapping the Getty logo on a work claiming "fair use" and then releasing the work under public domain would be a case of misrepresentation, because Getty still has a copyright claim on your work. Regardless of the copyright status, it's still a clear trademark violation to me.