Actually the BoB was too soon for the A6M to be of much use. The initial per-production run of aircraft did not arrive until July of 1940, with squadron production copies not reaching the field until early September (total production through Sept 30, 1940 was under 50 airframes), while the BoB was over by the end of October 1940.
The A6M was actually very much a flawed aircraft. It built its rep against aircraft that were already obsolescent, if not outright obsolete like the F2A, & I-15 (it is worth considering that the A6M actually had a negative exchange ratio vs. the F4F, as document by Lundstrom in his seminal The First Team, and no one ever mistook the Wildcat for an air superiority fighter). While it was an extraordinarily maneuverable aircraft, that was at low speed (at 200 mph it was "light as a feather" at 225 that went away), at higher speed it had considerable difficultly turning with even the P-40, much less the P-39, although it took USAAF pilots a while to figure out that it was a REALLY bad idea to get into a low speed turning fight with a Zero. It was also at its best as medium altitude, not the higher altitude favored in the BoB (this is also why the U.S. contemporary designs, including the P-39 and P-40 were of little help to the RAF, they were very good at 10-15K, much less so over 20k). The weakness of the A6M at altitude went well beyond just the limited oxygen supply (although increasing the supply by adding bottles or increasing the size of the existing supply bottles would add weight, something that quickly turns the Zero from nimble to, well, not). If the A6M's radio was not removed by the pilots (who preferred to lose the radio to save weight) and given even minimal defensive equipment it was a very ordinary aircraft since the added weight robbed range, speed and maneuverability. The aircraft, in its original form, was also severely undergunned, with two rifle caliber mg and two 20mm cannon with very small ammunition drums (60 rounds initially). This was later somewhat corrected, with the cannon ammo per gun increased to 100 (later 120) rounds and the replacement of first one, and then both 7.7mm machine guns with 13.2mm (.51 cal) guns in the Model 52 variants.
Facing the Spitfire, and even the Hurricane if handled correctly, the A6M would be remembered as what it was, an extremely maneuverable medium altitude fighter that had near zero survivability built into the design and an excellent place for good pilots to die.