It is explicitly stated by Stamets that the spore drive technology was seized by Star Fleet after the war started, against his will. So there is direct evidence the spore drive came after declaration of war. Under normal circumstances all that's happened is a cold war has turned hot, even by 6 months in the war they should not already be at the stage of resorting to experimental weapons like Spore drives but they instantly start down that path.

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It also uses a biological element as fuel and its main function is to allow Federation ships to rapidly attack deep behind enemy lines which makes it an overtly offensive weapon in contrast to their normal defensive ones. Then the Glenn happens and the project just continues, that incident alone should be enough to see the whole thing called off.

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There is also no logical explanation of Lorca's command of Discovery without significant control being made over ship assignments.