I'm not sure about the wire you have; however they do make flux-core wire for stainless. Go to/call your local welding shop, or airgas, etc. They'll be able to fix you up with the proper flux-core wire for stainless. As countryham said, flux core's not the best for stainless; but it will make the metals stick. If you take a grinder to the welds after the fact then it shouldn't be too hard to make it more aesthetically pleasing. Make sure you practice on scrap before you take on your keg(s). Get your wire speed, heat, and technique down and you should be able to put it all together fairly well.
IMO, buying a TIG machine, tank, argon (I think is what TIG uses), connections, etc is definitely not cost effective if you're just going to be using it for your brewing purposes. Flux core will handle the work you're doing just fine (at a slight sacrifice to the appearance to the weld).
Countryham does bring up a good point about getting the couplers professionally TIG welded in for you if you're worried about getting real quality (good looking) work. I think they'd probably charge you about $40/keg for two couplers.