I was worried about drilling into stainless steel and a 3/16 rod as well. I did a net search and asked my friends for advice.
Pic one shows a jig to hold the pin. I ran a router with a vee bit down some scape timber and then cut a couple of bits off and screwed then to the front and rear. It holds the rod nicely in place.
Then...with the 4' long Rod Drilling jig to support the rod, I used low speed, a 3/32 bit instead of 1/16. Yes a cobalt bit is essential, some WD40 but not much as the heat wasn't too bad.
Oh yeah I filed a flat first, and made a tiny center starting point as I was worried about work hardening the steel. Then I broke two 1/16 bits which explains the 3/32 cobalt.
I used steady pressure, cleared the chips regularly and was very careful near the breakout point easing off the pressure. I also watched at the start and rotated the rod a little to keep a 90 degree hole. Oh and I captured the drill deep into the chuck.
Then it was easy.... Afterwards I filed an angle in the end to make it fit nicely as you see in pic 3
I did this over a couple of days and worried about drilling these two holes for a few more. SS is hard!