What size and type of screws are you using for pivots and hardware? With that answer someone can pin point what your looking for.
Also, what "machinery" are you using? If you are using a hand drill or just a drill press you better have something with a pilot or that cuts both hole and counterbore at once.
If you have a machine tool, or if you can rig up a set-up which holds the part rigidly in the same spot in a drill press then possibilities to fudge it open.
If you can clamp the part in a vise, then clamp the vise to the drill press table lined up right over the spot you need the co'bore you could...
1. Drill the hole
2. Drill the co'bore with a drill, taking the tip of said drill very slightly less deep than where you want the flat bottom of the co'bore
3. Take a drill of the same size which you have ground flat on the cutting end and use it to make the bottom of your co'bore flat.
You can clamp the part up in any way to where it will not move at all and you are just changing tools.
A flat bottom drill will not cut well and will walk all over the place if you use just it. It will cut well enough to cut the drill point out of the bottom of your co'bore and the existing hole will pilot it. If you want to get fancy, rough the co'bore with a drill about .01" less than the diameter of the flat bottom drill rather than one the same size.
("flat bottom drills they make the rockin' world go 'round"...nemind)