As to whether it is intrinsically immoral then I don’t think the evidence goes so far.
I think it was a disciplinary measure that was instituted by the Church during a time in which the Church was still in a heavily pagan environment. Many ancient pagan societies considered tattooing a form of worship or dedication to false gods.
The discipline forbidding tattoos fell into disregard historically. Many saints are said to have had tattoos, especially in Western Europe, without censure from the Church.