"The heat might accidentally cause the gunpowder to ignite and the bullet could then kill you or someone else near you. "
I assume that you are not familiar with firearms and/or ammunition.
First off, there isn't gunpowder in modern ammunition. It's smokeless propellent, and it is far less sensitive to heat that gunpowder. Second, it would take substantial heat to set it off, or to set off the primer. This has been tested exhaustively both by the military (see Col. Hatcher's Notebook) and by the munitions industries.
Second, a projectile is propelled out of a firearm when the expanding gasses from the burning propellent are kept contained (and thus under extreme pressure) in the chamber and barrel of a firearm. When ammunition is cooked off outside of a firearm, there is zero chance of serious injury. The cartridge case remains intact, the gas dissipates in all directions and the projectile receives very little of the resulting energy. Again, this has been closely studied, not the least by the insurance industry.
I've tested it too. I've thrown live rounds into an exceedingly hot burn barrel. It can take 10 seconds or more for a round to cook off, surrounded by a fire that is over 1000 degrees.
All that said, I'm going to look around for the dummy rounds that I've loaded, and use one of those instead.