From the RAF's perspective, no. At the point that the first raid on Berlin was mounted the situation was not critical for Fighter Command (see e.g. the Air Situation section of War Cabinet Weekly Resume No. 51, August 15th - August 22nd: "... the main weight of attack was directed against our aerodromes (...) No significant success was achieved.") Tommy Elmhirst, at Air Intelligence in the Air Ministry, believed a raid on Berlin would result in retaliatory raids on London and certainly was not in favour, that view conveyed by the Deputy Chief of the Air Staff to Churchill who was not swayed in his desire to launch the attack (James Holland, The Battle of Britain).

Fighter Command was coping, but gradually being whittled away, particularly its trained pilots. Dowding was deeply concerned in the first week of September, discontinuing his policy of wholesale squadron rotation on September 8th in favour of Category A, B and C squadrons, stripping experienced pilots from quieter sectors to keep 11 Group in the South East up to strength, but by that point the bombing decisions had been made. It's also often overlooked that Luftwaffe fighter squadrons were under as great, if not greater, stress as their RAF counterparts.

There was a point that Bomber Command's C-in-C advocated attacks on Germany to provoke retaliatory strikes against Britain, but that was in May 1940 with the Battle of France going badly. The hope was that Luftwaffe fighters would be withdrawn to protect Germany and bombers would attack England, where they could more easily be shot down, though neither transpired (Peter Gray, The Strategic Leadership and Direction of the Royal Air Force Strategic Air Offensive Against Germany From Inception to 1945).