All this guessing at how a song will fit is funny to me and perhaps to others who write song parodies because we can find clever ways to work lyrics into the rhythm of a song. For example: Take the "Theme from Friends (I'll Be There For You)":
I choose you, Pikachu
I'll be there for you...
Another Tattoo could be a "Hey Soul Sister" parody for all we know.
Hey Soul Sister
While the original song may extend a word by holding "heeeey", you can fit two or three syllables into it comfortably. It might be awkward to conceptualize, but hearing it performed might make sense.
So based on that logic my new guess for "Another Tattoo" is "Tricky" by Run DMC. There we go!