You're gonna need a very thin oil to cut stainless steel weld. That stuff is beyond "wicked hahd".
Cobalt steel drill (nothing else seems to do it) and drill slowly with lots of pressure to get the drill to "bite". High speed on the drill just creates excess heat which anneals the drill tip and almost instantly dulls it while polishing the metal you're drilling so the drill can't "bite". Be very careful - it's easy to break off the drill using this technique.
Cooling and lubrication are key, too, but use a thin cutting lube, like less than 5 wt. I've had OK luck with 3-in-1 oil, but the best stuff I've found is sewing machine oil - about 1wt and clear - it drips like water. Walmart fabric department - Singer/Brother/Husqvarna all work well.
I had to drill through 1/4" stainless plate for my driving light mounts and went through two sets of Home Depot regular drills before I heard about the Cobalt Steel drills and oil from a guy who makes safety railings for sail boats. Old Navy CB who knew his stuff, but what he also said was "Always drill first and THEN weld Stainless."