It's an image. There's simply no way to add a watermark to an image that's both imperceptible to the user and non-trivial to remove. You'd have to pick one of those options.

I'm not sure that's correct. I'm not an expert, but there's a lot of literature on digital watermarks that are robust to manipulation.

It may be easier if you have an oracle on your end to say "yes, this image has/does not have the watermark," which could be the case for some proposed implementations of an AI watermark. (Often the use-case for digital watermarks assumes that the watermarker keeps the evaluation tool secret - this lets them find, e.g, people who leak early screenings of movies.)

That is patently false.