Something many of us are not a fan of, myself included, is when a videogame character is attributed with an excessively large inventory of weapons and equipment on their person at all times even though it would appear to violate realism, often by the universe's own rules. It's a common enough trope that we tend to disregard as mere mechanics of the game and not really something that's deliberate, especially when there is no provided in-universe explanation (i.e. the character is seen holding only one weapon in a cutscene, but during gameplay he's holding like 20 on top of ammunition for each of them).

However, in BJ's case, there is precedent to suggest he does in fact carry most if not all available weapons in the game despite not having a reasonable means to do so.

In Wolfenstein: The New Order, BJ is attacked by a robot which takes his Laserkraftwerk (which he was holding in the previous cutscene just seconds before and nothing else) and he retaliates with an assault rifle that he wields in his left hand, only to be punched in the face, which causes him to lose the remainder of his weapons except his knives and grenades.

Then in Wolfenstein: The Old Blood, BJ has to assume a disguise as a waiter and as such must relinquish his weapons, which include a pistol, an assault rifle, a shotgun, a handheld RPG, and a sniper rifle. This is despite the fact that prior to this he's seen shirtless and completely unarmed. And even though he's unarmed there, he canonically must be holding the lead pipe which he carries throughout the game's entirety.

The only conclusion to be made in my opinion is that BJ somehow manages to carry several weapons at once and use them effectively in combat even though we never outright get an explanation as to why.