I must be doing something wrong with the punch because even after the top was removed (by drilling) I couldn't punch out the rest.
In any case, I've managed to drill through pretty much the whole thing and will be done with that in another 15 minutes or so, once I have some time to myself.
Now my dilemma is what to use to fill the hole! The pin in there looks like a 5/32", but they don't make stainless bolts that small. I can't drill it up to 1/4" without destroying the hinge. Any suggestions? I considered getting a drill rod, cutting it to length, and bending the top... but I suppose maybe a thicker nail would do just as well. Any help is appreciated
I finished getting the old pin out tonight. For now, I just slipped a brass nail through. I doubt that will be my permanent solution, but it works good enough for now that I should be able to make some BBQ again. Thanks everyone for the help!