Everyone's talking about Edelgard starting a war, but after re-playing AM I just realised something - the war between the Kingdom + Church and the Empire is inevitable.
Dimitri has been investigating Arundel before Edelgard declares war. He's onto Arundel and he would very likely manage to connect Arundel to the Tragedy of Duscur if his investigation wasn't brought to a halt because of the war, which means the entire kingdom will be declaring war on the empire and demand Edelgard hand over Arundel to them. The Church will likely join in because they are allies and may even discover Arundel's true identity and through the empire, discover the activity of TWSITD in the empire.
War was bound to come. Edelgard just sped things up.