I also want to know the mechanism for the increased lead levels. Are kids chewing on the bullets? Is lead somehow deposited in the home?
I think you can interpolate what's happening from this part:
Firing a gun leads to lead particles on clothing that come back home.
So it sounds to me, like when you fire a gun, a few things happen. The bullet travels down the barrel and as it does, it makes some contact with the barrel, some particles of lead scrape or spald off. These particles are largely expelled by the barrel exhaust, but some of it can be deposited in or on the gun. So any gun that has been fired many times becomes a source of lead particulates that can get into the air. I'd expect a gun safe provides the same amount of protection from this exposure as a zip lock bag would. (But i'd still recommend using the gun safe). Theoretically, you could also reduce this exposure by thoroughly cleaning the gun before bringing it into the home, though I have no idea what level of cleaning would be necessary to achieve this, it could be a lot.