SUCCESS!!!!!!

I bought a pneumatic palm nailer (I had been looking at them and the battery ones for a while, but didn't think they would be useful - WRONG) and put a metal dowel in the threaded hole of the bracket. With this setup and progressively longer dowels, I was able to knock the bolt through. I love solutions where I get a new tool as part of the deal.

Before I reuse the bolt, I'm going to use my Dremel to grind the bolt and clean it up, so it will slide back in without binding.

Thank you to everyone who replied to my post. I hope I can return the favor someday.- David