Empires don’t have to win small wars. They simply need to maintain or return the situation to the status quo (i.e.; confine an emerging conflict, attrit to force a negotiation, withdraw, throw the weak sister regime under the bus to the enemy/foe, and then do business with the dominant new group who inherits the same crap that the old ally held).
Only rarely does an empire need to fight an existential war. The outcomes can be unpredictable and lead to the loss of the thing the empire wants to save. Better to engage in the first example above.