Straight Edge draws it roots from a song by the punk band Minor Threat, whose lead singer actually has a Coca-Cola logo tattooed on his arm (Punk's pepsi insignia is said to be a tribute). Considering it's underground roots, and the band that coined the lifestyle and phrase, I'd say that tattoos epitomize the lifestyle.
I don't see what tattoos have to do with addiction to promiscuous sex, recreational drugs, alcohol, or cigarettes. Unless you're addicted to inking up, then I don't see how you could be considered a hypocrite for calling yourself straight edge and having tattoos.
And by the way, it's not a gimmick. Back in his indy run, he actually cracked his skull during a bout and refused pain killers because he said it went against his chaste lifestyle choice.