Howdy Bill. I think I skipped this one. I believe this is what ITHURTZ was talking about when a cobalt bit was mentioned. This is not a cobalt but a high speed which is what I generally prefer, of those two. But its the flat-nosed variety, with a little pilot point that likes to snap off. Cobalt ones are black and harder but won't take the heat too well. Either would work for this. So there are several ways to remove welds, but the Blairs are my favorite. They work as fast as if the part was bolted on.
Matt, I also forgot to mention that in this case its probably fine to drill your pilot hole all the way through. Makes centering so much easier.
...AND ... that if all else fails, you can use a regular drill bit or better yet a stepped bit. You would drill away both layers of the whole weld spots then make new plug weld holes right next to the old ones. Just a thought, I'm sure you're ready to get past this so the car can be mobile again.