What makes it a BBGN and not a BCGN? I think it's at least partly political, if we're talking about a post-1945 ship. Something like how the Invincible class were "through-deck cruisers" and not "aircraft carriers", the
Kirovs might have been "battlecruisers" to avoid too much saber-rattling of the kind that might cause the US to say something like "you call that a nuclear-powered guided-missile battleship?"
Since this has to be post-1945, I think we can rule out 16 or 18 or 21 inch guns as a defining characteristic of a battleship. Instead, it needs missiles -- one of which can contain more explosive force than every shell every battleship ever fired, if you want to fit them with great big late-60s 20 MT warheads. That leaves armor -- sufficient armor to survive one or more conventional missile hits. That would be highly advantageous in any real-world scenario, such as a US "observer" ship being hit by SS-N-2s off the coast of Israel, as
@steamboy suggested. But, politically, it would be easy to say that nuclear-era battleships don't even need armor. US Steel would disagree, though, and it's possible our BBGN would be
very heavily armored. Plus, a ship double or triple the size of USS
Long Beach (CGN-9, 15,500 tons) could be called a battleship even if
Kirov (28,000 tons) is a battlecruiser and the "old", "heavy" battleships, the
Iowas, were 60,000 tons. Ultimately, the definition of a battleship is up to a country that has battleships.
I'll suggest that the difference between a BBGN and a BCGN is, in addition to what its navy calls it, strategic nuclear weapons. Ballistic ones, not Regulus. Perhaps it'd be a stopgap measure -- refitting the Iowas with ballistic missiles and nuclear reactors, until enough submarines could be built.
Long Beach was initially intended to carry Polaris.