Just a few thoughts...
Hitler did not order attacks on London, Hitler approved attacks on London after Goering asked if they were allowed to move to the next phase of attacks on commerce, ports and transport infrastructure.
Luftwaffe pilots and crews were not as seriously pushed as you'd imagine. Yes, there were some who were hard pushed but the majority of fighter pilots flew no more than 2 war flights a day and had at least 2 days off a week according to intelligence gathered from captured pilots. As for bomber crews, it was rare that they used more than a 1/3 of available aircraft in any 24 hour period. As I've mentioned there were exceptions to this but in general they were nowhere near as hard pushed as people imagine.
The whole myth of the RAF moving north to preserve their strength is just that, a myth. To prevent attacks on the ground Fighter Command would have to move northwards of Duxford and would therefore be conceding any defence at all of the skies in the South East and London. Effectively the Luftwaffe would have won and could bomb indiscriminately any targets in that region. Even if it was just a small scale retreat retaining Northolt, North Weald, Debden and Duxford the limited communications network and shorter interception times would cut the British defence capabilities by a half at best.
Statistically Fighter Command was losing a higher % of trained pilots than the Luftwaffe Fighter arm even with the Battle over British soil. From memory, on the British side the pilots had a 55% chance of returning to the flight of their aircraft was lost or damaged. This drops down to about 35% for German Fighter pilots, but the Germans had less than 50% the number of Bf109s lost/damaged during the battle than the British did.
There was a lot more going on in early September than you would think. For starters British airfields were starting to show sign of the battle, at Biggin Hill for instance the station commander set charges and ordered the demolition of the main buildings if another raid occurred, it didn't luckily. Secondly, Dowding introduced the Stabilisation scheme on 7th September effectively reducing the available front line strength of his command to less than 600 pilots. Thirdly, the number of aircraft available for immediate reinforcement had dropped to just 2-3 days in the first third of September.
Would a continuation of the campaign against airfields have had much effect? Yes it probably would in the short term. Would it have been a battle winner? Nope, not likely. As it was the Germans needed to advance their plans to the next stage to fit the timescale of invasion. They probably had about ten days of intensive flying left to them in September due to weather etc. and they chose to use that time attacking commercial docks, transport links and targets other than airfields as per their predetermined plan. Those ten days would not have been enough to destroy Fighter Command, weaken them yes, but not enough.