An oversized Spinal Battery should take up 10 systems, not 9 (
Spaceships follows a 1-3-10 progression). As a normal Spinal Battery requires at least one Core system, so should this, and it wouldn't be out of the question to require both Core systems be used up by the Spinal Battery. Looking at the
Wookiepedia entry, the death star actually had a diameter of 100
miles (160 km), putting it at SM+32 as a sphere. An SM+33 Spinal Battery would be 3 ZJ. As a laser (as it's described) or particle beam (most tend to treat SW blasters as particle beams), it would deal 3dx1,000,000 dDam; as a plasma weapon (which SW blasters resemble more closely) it would deal 6dx1,000,000 dDam. Unfortunately, both of these fall far short of what would really be needed to blow up Alderaan - ZJ is 10^21 J, while according to
Atomic Rockets' Boom Table blowing up Earth requires around 6x10^31 J, and the Death Star's maximum output was actually around 10^38 J. SI doesn't even
have prefixes that go that high! Even using the Cosmic Power switch doesn't help here, as that only multiplies output by 1000. Without Cosmic Power, the SS Death Star is on the same level as the last time Yellowstone erupted. With it, the SS Death Star is on the same level as the Wilkes Land Crater. Earth has tanked both of those just fine.
To get the Death Star's performance at its size, you'd need some new weapon. To blow up an Earth-sized planet to the same extent as Alderaan, you'd need to deal around 6dx1,000,000,000 (6dx2B for plasma) dDam. To match the maximum output of the Death Star, you'd need to deal around 2dx500,000,000,000 (2dx1T for plasma) dDam. That calls for either x2000 damage or x1,000,000 damage. GURPS rules are a bit more lenient - with Earth having 200,000,000 dHP (and ignoring its DR) based on its mass, Total Destruction can be obtained with "only" 2,200,000,000 dDam, which on average requires a bit over 6dx100,000,000 dDam (30,000 YJ for a laser, 3,000 YJ for a plasma weapon). That's achievable, if we use Cosmic Power (x1000 ouput, so 3 ZJ becomes 3 YJ) and scale your Death Star up to SM+40, for a diameter of around 2,500 miles. That's around the size of our own moon (2,169 mile diameter).
As a side note, I think setting it as an oversized spinal battery is probably correct. The Wookiepedia entry notes that all the weapons (other than the superlaser), habitats, etc are located in a relatively thin outer shell, with the innards taken up by the superlaser, reactor, sublight engines, and hyperdrive.