I used to do a fair amount of bullet casting, and the research I did on lead poisoning at the time turned up some results contrary to the popular belief.
First of all, elemental lead is not particularly dangerous to handle. It is not absorbed through the skin, nor through the digestive tract, nor by the respiratory system. There are large numbers of people who were shot in war and still carry bits of lead without ill effect.
The primary hazard of working with molten lead is burns, not lead vapor. The vapors that rise off the top of a lead pot do not contain significant amounts of lead - it just isn't hot enough.
The two things that are hazardous are impurites and lead salts.
Lead recovered from lead-acid batteries is contaminated with cadmium (which is toxic) and sulfer compounds (which are just plain nasty) including lead sulfate. Leave battery recycling to the professionals. Some lead alloys may contain arsenic.
Lead salts, such as lead nitrate, lead sulfate, and lead acetate are absorbed by the digestive tract and cause lead poisoning. Small amounts are not usually problematic, but lead poisoning is cumulative, and especially harmful to young children. Lead based paint is tightly regulated because of the link between eating lead paint and mental retardation in children. Children like to eat the flakes of paint because the lead salts taste sweet.
The myth of ancient Rome being poisoned by their plumbing is just a myth. If anything, they were poisoned by their use of lead acetate (aka "sugar of lead") as a sweetner in their food.
A bullet from the civil war might contain impurities, but is probably as close to elemental lead as they could get. Firearm technology at the time used a soft (pure) lead ball rather than one from a harder alloy. The ball has probably oxidized over time - is it covered with a layer of whitish residue? If so, you don't want to ingest that. I'd wash my hands after handling it.
Why not cast it in clear resin, or coat it in some other durable sealant (such as polyurethane)? Not only would you feel safer, but you'd be much less likely to damage the surface while drilling.
Oh, and for drilling - if all you want is a small hole for an eyelet screw, you could probably do that by hand with a small bit in a pin chuck. Lead is very soft.
Regards,
Eric