You could do the entire thing with the MIG.

Why are people obsessed with TIG ?

For an exhaust, TIG is pretty and nothing more.

If you want it to look all nice and pretty, by getting it professionally TIG'd together. There is no point tacking it with mild steel wire with your MIG. Unless they intend to grind the mild steel back out again.

You'll need to buy a roll of stainless MIG wire. But if you can weld, may as well just MIG the entire thing that way.