This is really a question about the chemistry of elements, particularly lead. In a bullet, you re dealing with either pure lead or lead mixed with antimony and tin; an alloy. In both of these cases, the lead is not molecularly bonded with another element. The way lead will react in this form is different than when lead is actually BONDED to another element.
Lead exposure from primers or old lead based paint is very different. The lead is attached to other nonmetal elements. In this form, the lead is an ion in SALT formation. The lead will come apart from the compound easier and then reattach to other things easier, thus it's absorbed by the body more readily. So eating a bullet is different than eating lead paint chips. The lead in the paint chips "disassociates" more readily and the person absorbs a much higher concentration. There is a very very small amour of lead that may disassociate from the bullet during digestion, but no like the ionized lead in a salt.
When shooting, the lead exposure is primarily from the primer compound. Most primers are the compound lead styphenate. This compound is a salt, so the lead is in ionized form and becomes aerated during deflagration. Inadequate ventilation in an indoor range exposes you to a higher concentration of the ionized lead.
Your biggest danger from lead bullets and shot is breaking a tooth. I don't recommend eating bullets..., but the actual lead absorption danger is minimal. Clean your game well and you and your kids are fine. I'm not a fan of solid copper bullets nor steel shot. They perpetuate ignorance and infer a problem at is nonexistent. Ive found both to shoot less accurate and both to wear out firearms faster. In some cases, they are just dangerous to the shooter and firearm.
Opinions vary on these bullets, but the chemistry doesn't.