I have a TIG and spot welder in my shop, but honestly I would just epoxy the parts back together if I were in your shoes.
The risk of burning through is just too great for thin metal parts like this (i.e. making a hole in metal). The problem with a welding shop is they may put the flunky/trainee on the job because if it fails it is not a big deal. Overheating Stainless steel is really easy when welding.
There is so much surface area on these parts it would be very easy to get an excellent glue job with the cheapest of wal-mart epoxies. As Vid said, sand the surface, clean with alcohol or other degreaser and glue them together. You will use a $1 worth of glue and half and hour of your time to do the job well. It will take you at least that long and more money to go to a welding shop and get them to do the job.
Most epoxies have strength in at least the 100's of PSI. Start to add up the square inches on that lockdown bar. That glue joint if well done would need 1000's of pounds of force to break it apart.