You aren't really talking about batteries here.
You're describing Supercapacitors; that is, capacitors with high power density. I would conceive of a supercapacitor as a device that must be recharged on a ship, and go through several discharge cycles, with astonishingly short reload times, before needing to return to the ship to recharge.
Super capacitors never have the energy density of proper batteries, and never achieve the potential difference of proper capacitors. They're currently a hot topic of research as a technology for helping to bridge energy generation\storage challenges for solar, wind, and electric car technologies.
A great concept for beam fighters.