Germany lost BOB because
1/Radar
2/Short range for Me109
3/No LR fighters(good)
4/No Strategic Bombers
5/Bombers they had not suited for not having air superiority
6/Changing plans midway from targeting RAF/Radar to industry
7/UK outproducing germany in fighter production
The problem wasnt the range of the bf109 but rather the poor tactical decisions made by the generals. The BF109's circled a long time at the french coast because they had to wait for the bombers which they were forced to escort closely, rather than climbing up and covering them from above to use their energy to their advantage. The BF109 and Spitfire were pretty simmilar in performance with 2 main differences. The BF109 was better in "boom-n-zooming" (diving on the enemy, strafing them and use the energy to get out of reach and up again) and they had fuel injectors, which meant they could pull much harder maneuvers without the engine stalling, which was a common issue with the early Spitfires from high G's, which didnt had injectors.
That had 2 main drawbacks:
1: The planes baislcy were out of fuel if they wanted to actually get back home when they reached the battle and very shortly after reaching it had to head back to base again, making them easier prey for the british pilots.
2: The british could easiely intercept the germans, which basicly announced their path of attack 20 minutes in advance to the british because of their radars. They could easiely get into position above the germans, which again were forced to stick with the bombers, and drop on them, giving them a massive advantage.
If the german high command had give the pilots alot more freedom in their tactical decisions, the whole battle could have went alot different for germany. They basicly didnt allow the BF109 to utilize their advantages over the british planes, which already had a huge advantage due to their radar to beginn with and announced well ahead of time where they are attacking from to the british, which could then get into perfect position to intercept them.