As you say, it has been cobbled up and the barrel doesn't seem to be the one that came with the gun.
If that black mark is in line with the face of the breech plug, the first thing to do is to look down the bore to see if the plug already has a smaller than bore size hole drilled into its face. If it does, get something smaller that will fit into the hole and see what the depth of it is. It might be deep enough to reach the center of the pan.
If it does exist and doesn't reach the center of the pan, try to determine what size it is. Then, get a "aircraft length" drill bit that is that size or slightly smaller. These aircraft drill bits are long enough to reach the breech plug in a pistol length barrel.
If the barrel length is short enough you might be able to use a standard length drill bit. My standard 5/16" bit measures 4 1/4" long.
Assuming the breech plug doesn't have a hole drilled into its face, you talk like you are planning to drill one in and, I agree with your plan. The biggest problem is keeping the smaller than bore size bit on center so it won't gouge the bore.
Wrapping the outside of the drill in the opposite direction than the drill rotates using some .010 thick brass sheet metal might keep the bit centered. It has to be wrapped in the opposite direction so it doesn't try to unwind when the drill bit is turning.
As for size, I think I would try a 5/16" diameter bit.
You won't be drilling much more than 3/8" deep with it so the volume of the drilled hole will be slightly more than 0.029 cubic inches.
That would be the size for 7.4 grain powder charge. You didn't say what the caliber is but if it is a .45, the powder charge should be somewhere between 15 and 30 grains so there is little chance of the ball bottoming out on the breech plug face when the gun is loaded.