I know a woman who injured herself deliberately, called the cops, and accused her husband when they arrived. He had just gotten home from work and had no idea what was happening.
So yes, a woman getting a tattoo to tell a false tale is entirely possible.
I’m thinking a lot of mental illness was involved in that story, so getting a tattoo is nothing.
I think there’s money to be made in safe inks/dyes that people could use to draw tattoos on their bodies as a sort of “try before you buy” or “I want to wear one but not have one” approach to tattoos. They would be the sorts of things that eventually wear off through time and bathing. Like dying the skin the way people dye their hair.
I’m aware of stickers and transfers that amount to as much. Also actual “sleeves” you can wear and remove that is a fabric that mimics the color of various skin tones with tattoos printed on them.
Probably, such safe inks/dyes already exist. But I’ve not heard of such used as alternative to permanent tattoos.