Can't loot Rivet Gun from first Rosie in BioShock Remastered

I have four weapons equipped at the moment: wrench, pistol, machine gun, and shotgun. Now I've killed the first Rosie Big Daddy, and it dropped its Rivet Gun. When I point at the Rivet Gun as in the screenshot its name is displayed suggesting it's a recognized object that might possibly be looted, and from what I've read that should be the case.
However, when I press E (or any other key) nothing seems to happen, as if I can't interact with this object at all.
What's going on? Can I actually loot this item now? Do I have too many (4) weapons equipped at the same time? Can I drop some to pick up this one?
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Can you use the rivet gun in Bioshock 1?
Unlike the Drill, the Rivet Gun is a suitable weapon for any player, regardless of playing style. At the start of the game, it can be used to help determine one's combat preferences, and will remain a useful tool in one's arsenal for almost any situation afterwards.How many guns are in Bioshock?
Twelve of the fourteen weapons in the game can be upgraded at the Minuteman's Armory vending machines, up to four times.Can - Mother Sky (1970) [HQ]
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