Agile fixed-wing aircraft integrate the efficient, high-speed capabilities of conventional fixed-wing platforms with the extreme maneuverability of rotorcraft. This work presents a nonlinear control strategy that harnesses these capabilities to enable autonomous flight through aggressive, time-constrained, three-dimensional trajectories. The cascaded control structure consists of two parts; an inner attitude control loop developed on the Special Orthornormal group that avoids singularities commonly associated with other parametrizations, and an outer position control loop that jointly determines the thrust command and attitude references by implementing a novel vector-projection algorithm. The objective of the algorithm is to decouple roll from the reference attitude to ensure that thrust and lift forces can always be pointed such that position errors converge to zero. The proposed control system represents a single, unified solution that remains effective throughout the aircraft's flight envelope, including aerobatic operation. Controller performance is verified through simulations and experimental flight tests; results show the unified control scheme is capable of performing a wide range of operations that would normally require multiple, single-purpose controllers, and their associated switching logic.