I've finally discovered how the tattoo button disappears! This, however, leads to other bugs related to it.
But first things first. For me the tattoo button in the inventory disappears whenever I open the inventory via the Inventory button in the Mansion's main menu, and as soon as I used this button even once, the tattoo button never shows up again until I shut the game down and load it up fresh.
The tattoo button only appears for me, when I access the inventory screen through going to a character's info screen and from there go to the inventory. However, accessing the inventory through this method makes the button always appear, regardless of whether or not I have build the beauty parlor. It appears both on the master and any servant, even if she/he has not trained the body modification loyalty upgrade. And using one of my edited saves, I can confirm that even when none of the prerequisites are met, I can apply tattoo's as long as I have the inks.
Anyway, I hope my observations can help other players and perhaps even the developers. Cheers and happy gaming everyone!