If you're referring to the JJ1 comic for the times when Jazz was missing the LFG 2000, also note that he was missing both the gun and the backpack in nearly all panels. Besides, he could easily have left it in the Jazz-01. Not to mention that on Page 6, he's wearing the backpack in one frame, but it's vanished in the next frame. Nick Stadler may be a great artist, but he's not too consistent.

If you compare the size of the gun to the size of the backpack in both the artwork and sprites, the gun is much larger. In the JJ2 sprites, if you set the gun vertically, it continues from his neck down to his feet (given his normal stance). I understand that sprites aren't always accurate to the artwork, but that's a pretty darn big difference.

And, going strictly by official sources, Jazz would carry his gun on a strap around his body. Check the frame of the JJ1 comic in which he is injured and Chuck flies away. Jazz has a strap there leading toward his gun. We know that's not a backpack strap because it goes around his body, not a shoulder. Furthermore, if you've got a JJ1 sprite sheet handy, you can see the strap on Jazz's shivering frames and on his "bored" animation. We know that these aren't backpack straps here because they are much looser than those straps as depicted in any other source. Although admittedly the only animations that actually show the backpack/ backpack straps are when Jazz leans on a wall and when he tumbles (barring the 3d segments of the game)

And really, it couldn't be the backpack, due to the fact that we see both at once in the JJ2 manual cover and on most of Jazz's JJ2 sprites.

All I'm really trying to say is that you can't hide a gun that big in a little backpack, and Jazz wouldn't have a problem with giving it up when his kids were in danger, whether it was his father's gun or not.