A declaration of war or an authorization of force are both Congressional approvals for use of military force against an enemy of the state. I get that's inconvenient for your argument, but they're the same.

And the president doesn't have the authority to declare war on his or her own accord, full stop, because the constitution explicitly gives that right to Congress (and no other branch).

Any convoluted timelines around requests are immaterial to those facts.

If the president uses the military to attack an enemy of the state, without Congressional approval, that's outside of his or her authority.