I thought there was a Starfleet regulation on the types of personal allowed to carry weapons while on duty aboard a Station/Ship. I do know that according to the old technical manuals the phaser used by the helmsman in Farpoint was claimed to have been stored in a compartment near the console, even though he clearly draws it.
And that weapon caches under security seals are located throughout starships so that they can be used as needed, those caches seem to more or less be as frequent as damage control and first aide lockers when you see them used in TNG.
I would assume that with weaponry being relatively “on hand” aka use a Determination and the trait “Always Prepared” to quickly open a locker and grab a Type-1 Phaser is fair game. But just carrying around weaponry is probably against starfleet regulation, especially as (unlike the current series would have you believe) Starfleet principally reaches for Diplomacy first not a gun.
I’d probably talk to your GM and the other players and explain why you think that character is acting out of starfleet tradition. The game rules has room to make it part of the storyline, perhaps they get demerits for their aggressive behavior and should see a the Ships counselor. Or if your the captain make it a Standing Order.
You could always point to a character like Tasha Yar. She didn’t “always” carry a Phaser when on the Enterprise, certainly when not on duty. And her backstory would be the sort to make someone want to always keep a weapon on hand.