That was addressed way back in the thread, MW. They keep records of the pre-change genetic profile as well, if only so that gene-tweaks (on-the-books ones like you're talking about, at least) can't be used to avoid DNA evidence linking someone to a crime. There's also recovery time, and plastic surgery leaves marks.

Marks that are in the main only detectable by in depth examination. Simple easy to see marks aren't really the thing for celebrities, and a doctor good enough to opperate on a NMS is good enough not to leave marks that're easily visible..
As to the records issue I'm not sure... if two people have the same DNA, you can't spot the genetweaking, and one has a SIN, who will you believe? (as to recovery time, there're a few ways around that, some of which I'll go into below.
But funny thing is, that facebook profile of yours doesn't mention anything about that month you spent in a clinic... isn't that odd? And of course, the next time you pass an S-K or Ares border or security checkpoint, they might want to do some additional controls and have their own medical team do some 'minor examinations'. All for your own and public safety, of course. "Don't like it? Tough. Maybe next time you'll have your work done at one of our clinics instead, huh?"

Alt scenario then; what if you got that info in advance, did the surgery beforehand to make the 'new' holder look like the original, and simply swapped the SIN? That's not detectable without a damn good medical scan, performed at the word of a SINless. And just because you're SINless doesn't mean you're a tramp. You could be a criminal with genetweaking trying to make trouble for an upstanding citizen!

Another idea with the first; claim that you're beginning surgery and recovery with the theft. Again, the authorities have to actively investigate to catch you, and do it on the word of someone without a SIN, and on what more or less amounts to their word.
Lose your SIN in the crash and then LIE. You don't want people to know that the SIN is well and truly gone, just misplaced. So lie and tell people it was stolen and you're trying to get it back. If people know you have no SIN, you're basically an animal in Seattle. But if they think you're a victim of identity theft, you're a person.
Hmm... perhaps, but for a NMS? They're animals anyway.

Those Frogs might give 'em SINs, but we don't do that stupid drek here! I ware my Naga skin boots and like 'em.
(Or wait, is that the CAS?)