I have a partial answer, from the Air Historical Branch's narrative account of the battle. This was written in 1943-44, so as to set down an account while memories were fresh and documents readily available. It was a primary source for the British official history of the battle, and many subsequent authors, but was not published in its own right until 2000.
It has the German figures for what they tried to drop for August, September and October, in Appendix 32, and in Appendix 17, the weight of bombs dropped on London for each day of September. The book reckons that the documents these are based on are reliable for German plans, but take no account of bombers shot down, bombs dropped in the sea, and so on. They are in metric tons.
- August: 2,548 tons of HE bombs, 165 tons of incendiaries.
- September: 8,909 tons of HE bombs, 429 tons of incendiaries.
- October: 9,057 tons of HE bombs, 253 tons of incendiaries.
Yes, this is quite a small scale of bombing compared to Allied bombing later in the war. The Luftwaffe really wasn't organised or equipped for strategic bombing. It is not clear why there are no figures for June or July, but there are references to parts of the records of the German Air Fleets being lost, which would have been the primary source for them.
If the figures for incendiaries seem low, that is likely because of an easy error to make when reading Luftwaffe statistics. They reported HE bombs in metric tons, and incendiaries in terms of "containers." Those held 36 incendiaries, each weighing 1kg. The number of containers of incendiaries is of the same order as the number of tons of HE bombs, and it is easy to mistake it for the number of tons of incendiaries. The number of containers for the three months in question is:
- August: 4,596.
- September: 11,926.
- October: 7,021.
Addendum: It seems very possible that the British did not try to compile detailed figures for bombs dropped or estimates of their weight. In a heavily bombed urban area, it can be very hard to distinguish the effects of several explosions. At the other end of the scale, many bombs fell in the countryside, doing no harm apart from craters in fields.
The British do not seem to have regarded the bombing on land during June and July as very significant. Richard Overy's The Battle of Britain, a short volume of insightful analysis, reckons that the main effect of those raids was to provide training for both sides' air forces. The British gained more, because they gained experience with the operation of the air defence system they had been training with, while the Germans were attempting to learn strategic bombing from scratch, having been previously trained as an army support force.
The Battle of Britain, T.C.G. James, Frank Cass Publishers, 2000. Crown Copyright.