When its called a War, It is meant in the sense that it was a great war between Light and Dark. Julius, the Belnades, Alucard, Morrises, the Church, and anyone involved in the fight against the supernatural, versus Dracula at the height of his power, and his army of monsters and undead.
It likely took place in eastern Europe, like all Castlevania games, in its usual setting.
Also where does it say that military involvement took place?
All i recall is Hammer being sent by the Military to investigate the floating castle in the sky. That is a natural response to a situation like that.
And yeah, the rivalry is silly and unnecessary. If the Morises still pass down the whip and train to be Vampire Hunters, then they are very well in touch with the Prophecy, and that it came to them from Belmont, and must eventually return to Belmont in 1999. They are the caretakers of the Whip, and Belmont substitutes. Not replacements. And will never be any more than that. It is their destiny to hold the whip until 1999.