You can enlarge the hole (drill it or file it as you prefer) then fill the hole with a solid plug, steel or bronze (rather than a pre-drilled bushing). Solder if you want, or champher edges of hole and peen in tightly if you prefer. Plug just needs to be big enough to have sufficient solid material around the new hole, and small enough to stay hidden under the cock. Non-round shapes (eg. square) will guarantee plug won't spin in hole during drilling.
Then mount the plate face down on a squared chunk of hardwood, mount the hardwood in the drillpress vise, etc., then screw on the bridle, drill a pilot hole by putting the drill through the small axle hole on the bridle. When pilot hole is complete, then remove the bridle and switch to a large drill to cut the large axle hole to size (or to slightly undersize if you want to ream to final dimension). Helpful if you use a large drill with a non-cutting pilot center, same size as the pilot hole. That guarantees that the large hole doesn't wander.
Back in the day, we can assume that the pilot hole would have been cut with a small straight-shanked spade bit. Such a bit won't cut the bridle laterally as you drill the pilot.
Lathe tools and drill-press type tools are shown in the Wyke's catalogue (printed from 1759-1770) which could have been used to accomplish such drilling 200 years ago. Most of those tools were operated by a bow drill or a brace. If you want to use a modern hand drill, rather than a drill press, then you may want to build a simple hardwood squaring jig to help you keep everything square. Hardwood squaring jig will also work with bow drill and or brace. Non-cutting pilot on the large drill will help considerably if you are going to do this with a hand tool.
