There is another thread recently on drilling stainless. See if you can find it, there was a lot of good info. The net of it is that if you have a sharp bit, good lubrication you will keep the heat down and successfully drill the hole. As soon as that bit dulls, you will heat up the work and if you get too hot, it will work harden. Once that happens, your basically in deep doo doo. You will be lucky to get the hole drilled at all and if you do get it drilled, it will cost you MANY expensive drill bits to do so.
The machine shop will have this drilled in moments. You will wonder why you pay so much. You are not paying the shop for their labor, you are paying for their knowlege. But, if you screw up and heat harden the stainless you will personally be at it for hours and hours and may not make it through at all and will easily be most of the way to that hundred bucks just in cobalt drill bits or titanim nitride.
Typically, you drill the first hole, no problem. You wonder what all the fuss is about. The second hole goes smoothly until the end when you don't break through when you think you should. You either get through, or about half way there suddenly your hole starts smoking the lube and you realize you are in trouble. If you put a new sharp bit on after it cools, you will usually get through, though just barely. If you start the third hole with your 2nd bit which is now seriously dull, you likely will never get that third hole drilled. Ever.
I've drilled many many holes in SS. I learned the hard way. Sharp bit. Lots of lubrication. Go slow. Walk away for an hour at the first signs of heat. Come back with a new or sharp bit. Be patient.