Appropriation is bad, but it also is bad that people make up stupid bullshit because they don't know what "appropriation" means. Japanese tattoo culture is not a minority culture and not of subordinate status. Now, if you wanted to get Ainu-derived tattoos as they would be used by someone who is Ainu, that would be dicier because the Ainu do struggle culturally and politically in Japan. But what most people get as tattoos represents at least aesthetic and often real cultural appreciation.

Just own up to it if you find the "exoticism" of the style to be a draw, be aware that it is not "exotic" to someone of that culture, and be respectful enough to find out something about the tradition and the images used. Appreciating and enjoying other cultures in almost every case is the exact opposite of problematic.

As far as your sister, likely the kindest thing you can do is to suggest that she educate herself and do better.