Stainless is a bitch to drill through!
I used almost exactly the same drip tray with back-splash as you and drilled 4 holes through it:
The stained chunk of wood that the shanks go through is used to move the taps away from the chalkboard a bit so that it's easier to grip the handles, but also serves to hide imperfections from drilling hand-held.
As others mentioned, I used a step drill bit to drill through the SS:

Just keep going down until you get the size you want. Take a break if things start to get too hot. Anything else is going to cause you lots of headaches. If you use a drill press too, even better! (Wish I had one)
Kal