Just to clarify, the Italians used license built copies of the DB601 and DB605, not the DB603. The 603 was used on large German bombers like the Do217 and 335, as well as on the Ta152C. It was tested on the Fiat G55 but never put into production on that aircraft.
On that note, the Fiat G55 was tested vs the FW 190A5 and Bf 109G4, and bested both of them, causing the German officers who were involved in those tests to recommend it replace both German fighters poste haste! (Milch, Petersen and Galland)
One of the design limitations of the Bf109 airframe was that it could not be modified to accept the larger DB603 engine, something the Ta152 and G55 could. Other things like limited armament configurations, high elevator forces at high speeds, cramped cockpit, poor vision, no capability to use improved 'bubble' vision canopy, high aoa on landing, and high skill/training level for effective or safe operation were also indicators of the 109 being at the full extent of it's development potential.