If its just a bit of trim, and its nothing special, id be inclined to use normal mig wire.
Is it a sliding fit over the old? In which case you could use self tappers or make a slip joint and use an exhaust clamp.
As we already mentioned, its not the right way, sometimes you need to weigh up if its really worth buying different wire and gas to do a job like that.
It might fail a month previous of if doing it with 309 wire, but hey, its just an exhaust trim. Its also not to be used if its something that could be dangerous if failiure occured.
I have done a fair bit of this on sculptures, using mild steel wire, nothing too structural and have had great results.
One important thing to think about is protecting the work from stray specks of spatter, they stick like hell to the stainless, and leave ugly rust specks, I think a piece of card or on tube a coke can would do the job, just enough for them to bounce off.
I only realised it was a problem a few weeks after the job.
Chunko'.