It's hard to tell but that looks like a pretty shallow hole for the job it has to do. I wonder if the bottom of the hole is really the top of the old bolt that may have broke off previously. Although the handlebar bracket is keyed and there's no side to side force on the bolt, it still has to hold it down firmly. One would think an 1"-1-1/2" deep would be adequate. I would find a drill bit that fits snug inside the threaded portion and drill just enough to get a center started then switch to a smaller bit for the 5/16" tap and make it at least 1" deep. If all else fails, your welding buddy can fill the hole with weld and you can re-drill it but your nice chrome will turn blue.