Indiana Jones and the Great Circle devs say an Indy game 'could never be a shooter, should never be a shooter,' so they're embracing his signature whip, improvised brawls, and disguise-based stealth..
Indiana Jones carries a revolver and occasionally uses it, but yes Indiana Jones isn't a story about a gunman. The way I would implement the gun mechanic in an Indiana Jones game is give him limited ammo, maybe 6 to 12 bullets, and few opportunities to restock. Indy should defeat enemies either by setting off traps/environmental kills, fisticuffs, or with his whip, with his gun as a weapon of last resort. Also, there should definitely be moments where the player feels like a genius for remembering he has a gun.
Don't do it randomly. Either design sequences where Indy is disarmed, or actually embrace the 6 bullets per level idea, don't puss out and leave bullets just before every conceivable thing Indy might shoot. Make every encounter doable with the whip/fists/environmental stuff, but make shooting something a very cool way to handle certain situations for players who conserve their ammo.