There are two rationales for a limited strike on Syria: (1) Deter Assad from using chemical weapons again and (2) Send a signal to Iran that the US will enforce a red line against nuclearization.
The implicit threat of a tough response from the principal if an agent takes a Bad action must be coupled with the promise that there will a soft action if he takes the Good action. If the principal always takes the tough action as he is a hawkish type or always takes the soft action as he is a dovish type, the agent’s incentives are not responsive to the principal’s strategy. In the absence of incentives, the Bad action is optimal for the agent. The principal’s welfare is low because the agent takes the Bad action. The principal has to signal he is a coordination type to truly change the agent’s incentives.
Voters don’t like war just for the sake of it – they pay the costs and see little benefit (Kantian peace). But they can be convinced to go to war if there is a serious national security threat. There are coordination types. By going to Congress, the President imports the preferences of the median voter. The agent knows that the Good action is more likely to be met with a soft response and a Bad action with a tough response. The agent responds is more likely respond with a Good action. So the President’s payoff goes up whatever his true type. Hence, going to Congress is a great idea for the President whether he is hawkish, dovish or coordination type.