Liam tells Variety -- "I just got a call to do a one day shoot on 'Hangover' as a tattooist in Thailand ... that's all I know about it" ... and the whole deal went through his "A-Team" pal, Bradley Cooper.
As we first reported, Mel Gibson was 86'd from the role -- in which he would have played a tattoo artist -- after friends of the cast put political pressure on the actors to raise hell.