Re: The Constellation's registry number
In "Return To Tomorrow" Kirk says they avoided destroying themselves wherein Sargon adds that his people also survived their primitive atomic age. None of that precludes a large scale conflict where nuclear weapons are used even minimally.
But that also doesn't preclude that several conflicts were fought either. The Eugenics Wars were part of World War III, but World War III ended up being referred to as such even though it was made up of many different conflicts in the same general era.
"I had no idea you were so... formidable. " - Anan 7 to James T. Kirk, A Taste of Armageddon