It might be a battery in a comic or novel but not in the films. But yeah that’s actually the medical kit. Jungle Hunter and the crucified Classic Predator both have the guns attached directly to the medical kit. The AVP Predators do this as well with the exception of Wolf who is attached directly to the armor like City Hunter.
As far as this game goes I don’t know why they just pull an injector from the back but it might’ve been for the sake of expediency in the animation. Though given it is a common misconception (which is weird because it’s been shown a few times on film) the devs might’ve been under the impression it was for a battery pack.
I don’t know where this came from either. I think it might’ve been from an AVP comic but I can’t place which if it did.