I wouldn't recommend traveling with a handgun, not across multiple states. You can think you're doing everything right and still get in some hot water.
I'm sure the poster is responsible, but having a loaded handgun (I assume it will be loaded, because you can't shoot a bear with an unloaded gun locked in a box) during a camping trip with 4 kids under 10 makes me nervous.
The old - treat every gun like it's loaded, and always keep it loaded because it doesn't do you any good if it's not, holds true for this situation.
I've been camping with my kids and nephews. Hard to keep track of them all and they are always digging through bags and gear - looking for smores, flashlights, ect. Hard environment to always know what's going on, and that's when accidents happen.
But, his concerns aren't unreasonable and if I was in a place with a heavy bear or cougar population with my kids, I'd probably want protection too. Just need to be extra cautious