I have twice now survived a thaumite infection: once by simply using the potion that cures all debuff, and another time by just healing through the damage until the debuff expired. Neither strategy unlocked the Honey Badger Don't Care achievement either in Steam or in-game. Both infections were caused by intentionally stepping on a thaumite nest trap. The achievement for DYING to a thaumite infection, however, works properly.
Finally got this to unlock on Elvishly Easy. It wasn't unlocking on Roguishly Hard for some reason. Unless it's just a bad trigger and the difficulty was just a coincidence.