Saw an interesting observation. It's possible that the gun here isn't actually loaded, and Jax knows that. The whole game was kicked off by Caine's trust exercise where Person A unloads the gun and Person B either shoots themselves or Person A. Considering that at the start of the episode, and that Jax sees himself as a wacky cartoon character pulling tricks on a cartoon show, he was probably trying to set up a bit to end the episode on. Which would probably go:
>Pomni pulls the trigger
>Doesn't fire
>He says some callback joke to the trust exercise
>Shoots her, joke made at her expense to end the episode on
But she's not on-script like he thinks he himself is, and it's making him internally pissed she's not playing along or "committing" to the bit. So by not going with the Exercise's option A or B choices (shoot him or herself) the show set up at the start of the episode, her choice to fight him instead kills the payoff and gives him no out, forcing him to be more human to defend himself which is what he's trying not to be.
Just a theory though for this one scene, gun could very well be loaded.