Okay, but can they sustain the pilot and aircrew losses this would entail? Because the Germans need more aircrews if they want to sustain a Battle of Britain (an airplane mix more weight towards bombers), and they're going to be suffering a far, far worse rate of irrecoverable aircrew losses (both sides planes are going down over Britain; for a British pilot, that means a nice farmer patching them up and sending them to the nearest military hospital; for any Germans, said farmer is going to either call in MPs if they're lucky or stab them with his pitchfork/Home Guard spear if they're unlucky).
I think you're overlooking something here - the German pilot is armed too, probably with a service pistol. Being younger, fitter, and a trained military professional, he is highly likely to win the initial encounter with the British farmer, thus giving him access to the farmer's equipment too. He will be able to use this, plus his training, to live off the land, taking what he needs from the civilian population and attacking any targets of opportunity he comes across. As the air war intensifies, dozens - perhaps even hundreds - of these ersatz Fallschirmjager will be roaming the British countryside, terrorising the civilian population and making military movements almost impossible. Neutralising each one will take a far greater number of personnel, who will no doubt take casualties in the process.
Meanwhile, the invasion force is assembling on the other side of the Channel. Unable to move forces to counter it, unable even to defend their civilian population, the British government will be forced to seek terms of surrender. Personally, I give them a month at the absolute most before public pressure becomes overwhelming.