Just my 2 cents, but take a deep breath. First of all, to repeat, INK ALLERGIES ARE VERY RARE. If you don't believe me do the research - in the medical literature where cases would be documented and confirmed, not in online forums where everyone who ever had an itchy tattoo said they were allergic to the ink.
Second, it sounds like ONE doctor, not doctorS, didn't know what it was.
If it's a simple infection, it should be easy to clear up.
Good luck.