It sounds like that guy believes you did something really unforgiveable to him, where's he's taken it super personally. I don't know the context, or the specifics of the conversation, so i'm very likely wrong. Either way, the guy should have kept his cool, and never should have lost his shit. Anybody working a gun club, gun store, or anyone holding an FFL in my opinion needs to be cool as a cucumber in any situation. I hold myself to that standard as well. Sure I get angry at people, but I sure as fuck don't lash out at them, refuse to let them leave, and threaten to shoot their dog. I would file a police report, that way it's official record he threatened to shoot your dog, and/or harm you, so if something does happen, the police have a official report to look back at for finding suspects.
I'm sure you've already decided to avoid the guy as much as you can, maybe get the word out to the other club members that the guy is nuts so he can be removed/replaced in the club. I'm sure nobody at the club wants a psycho with a firearm there.
I have a neighbor that i've had years of great conversations, and very calm, productive discussions about our opposing political views, and we've both thought of ways we can meet in the middle on various topics. Never had a problem, always enjoyed hanging out with him on friday or saturday evenings by a fire, watching starlink satellites go over, and just shootin' the shit.....until recently. Something has changed in him, and not for the better. He text me & my wife one evening, told us we were horrible people, and never to talk to him again. He said I was targeting him, and thought I was going to hurt him and steal his money because i'm a Trump supporter. So I screenshot his messages, deleted and blocked his number, and left it at that. We cross paths last september while the wife and I are walking and he acts like it never happened. I talked to another neighbor, and the crazy neighbor went down to his shop behind his house, walked in to the fridge, took a beer out, took a drink, then told him he was going to kill him and turn him into taco meat.....in front of his wife and stepdaughter.
The assessment that there is a lot of anger, fear, frustration, and just pent up mental illness, is real.
OP; be careful man, psycho's are super unpredictable.
Branden