To properly mig stainless you need to run straight Argon, SS wire, and reverse polarity. In your situation I’d just use a nut, bolt, and a couple washers.
Nope.
The only typical metal you MIG weld with straight 100% argon is aluminum. All steel MIG welding is done with some amount of active gas in the gas mix, whether that active gas is O2 or CO2.
Short-circuit transfer mode GMAW (aka MIG) on stainless steel is typically done with tri-mix or 98-2 argon-O2. Spray mode GMAW on stainless is typically done with 98-2 argon-O2.
GMAW polarity on stainless steel is the same as on plain steel, DCEP, DC electrode (wire) positive. Work clamp (aka 'ground') is the negative.
To weld stainless steel and keep it (mostly) stainless, you need the proper filler wire and the proper shielding gas and the proper procedures.
Yes you could sort-of weld two pieces of stainless steel together (maybe) with a HH-140 with C25 gas and plain steel filler wire (ER70S-6). The "maybe" is if the HH140 has enough amperage on 120V input power to properly melt the flange AND the broken-off stud back together. And the fact that using C25 gas and mild steel filler wire means that the weld is no longer stainless. And you might get enough carbon pick-up from the C25 in combination with the ER70 wire to possibly get a hardenable witches-brew of an alloy mix in the weld puddle that is hard and brittle.
To the OP, hassle or not just send it back and have them send you a non-broken exhaust. Or drill it out and replace the stud with a stainless steel bolt and nut.