If I were a player, I would be a lot more likely to want to build that überstarship if its name sounded dreadful and harmful. I don't know if by "a Death Star feeling" you mean that, or mean something really huge, both of them or something else.

To me, something related to fortress sounds huge, but not dreadful. On the other hand, Stellar Annihilator or Galaxy Destroyer sound like vessels I would really want to build.

Note that's just my preferences and I'm sure other people will show preference for names that evoke other things.

A space empire, you build one or more (but usually just one) super-something-combat-gigantic-vessel things.

The Empire built two death stars.

These class of objects are typically called Megastructures - just astro-engineered ones. I'd give them individual unique names just like boats, colonies, planets, and cities have names.

Because Megastructures take so much work to build, it's unlikely there'd be many identical-planned megastructures. How many identical skyscrapers do we have? Few, if any.

Even large boats are usually have only two or three "sister ships", and even then may have modifications unique to each one because new technology gets invented while they are constructed and each one takes so long to build. Boats do have "classifications" to help military quickly group ships mentally into different classes of firepower or different cargo-tonnage. These classes are usually named after one of the first ships (the "lead ship") of that new class. We have Nimitz-class aircraft carriers, and they are named after the Nimitz which was the first of that class.

Rather, each megastructures would be extremely unique and vary greatly in size, shape, blueprints, and so on. The first of a specific size would be given some unique name (i.e. the MS Wiggins), and then later ones of similar size would be called after the original's class (i.e. the MS Monolith might be a Wiggins-class dyson sphere).

The Empire built two death stars.

Which was only to be mass-produced by ripoff games.

If there exists a possibility that even though it is a super weapon, two copies may exist at the same point in time, I would go along the lines of:

{unique name} + class + {purpose}

Alezandersmith class Assault Starbase

and casually refer to it as the 'Alezandersmith'

(of course, the name should be more badass, like the Death Star, it should be tied to both space AND fear: Star and Death, respectively).