Cochineal Beetle

It lives in the Cavern layer and it is rare, so you do not hunt one down so much as run into it while doing something else. A Lifeform Analyzer is the only dependable way to know one is close, since it sits on the short list of things that accessory bothers to announce. If you specifically want red dye, wearing the analyzer during normal cave mining beats camping any particular spot.

It reads as scenery. A fat beetle shuffling along a cave floor looks like a snail or a worm, and plenty of early characters walk straight into one expecting nothing to happen. It is a real enemy with real contact damage, and at the point in a run where you are still exploring caves in starter gear that hurts. Deal with it at range the first time you meet one, or at least approach it the way you would approach anything else down there.

The payoff for killing it is the Red Husk it leaves, which exists to become Red Dye. That is the entire point of the creature. A small pile of husks covers a whole playthrough of dyeing for most people, so there is no reason to farm these beyond whatever you stumble across.

Cochineal is a real insect from Mexico and South America, farmed for centuries for the deep red pigment it produces, and still used to color food and cosmetics today. The real one is a scale insect rather than a beetle, so the game version has been given a promotion it did not earn.