It's doable.
Admittedly, the "SINner" negative quality specifies that the 10 BP version of the quality represents having a criminal SIN and a verifiable criminal record; It'd be kind of weird to have a criminal background without having the SIN to prove it.
That being said, the Erased positive quality represents having a friend who actively works to erase any criminal SIN you might acquire on an on-going basis.
It seems reasonable that this could also have been a one-off event in your character's past, allowing you to start the game with a criminal history and no criminal SIN or record.
Also, don't forget the possibility that your character could have had a criminal past but was never caught, or who managed to get off due to having very skilled lawyers.