So people were thinking in that direction.
Ok from this article:
https://en.wikipedia.org/wiki/Pulsejet
In 1934, Georg Madelung and Munich-based Paul Schmidt proposed to the German Air Ministry a "flying bomb" powered by Schmidt's pulsejet. Madelung co-invented the ribbon parachute, a device used to stabilise the V-1 in its terminal dive. Schmidt's prototype bomb failed to meet German Air Ministry specifications, especially owing to poor accuracy, range and high cost. The original Schmidt design had the pulsejet placed in a fuselage like a modern jet fighter, unlike the eventual V-1 which had the engine placed above the warhead and fuselage.
The Argus Company began work based on Schmidt's work. Other German manufacturers working on similar pulsejets and flying bombs were The Askania Company, Robert Lusser of Fieseler, Dr. Fritz Gosslau of Argus and the Siemens company, which were all combined to work on the V-1.[6]
With Schmidt now working for Argus, the pulsejet was perfected and was officially known by its RLM designation as the Argus As 109-014. The first unpowered drop occurred at Peenemünde on 28 October 1942 and the first powered flight on 10 December 1942.
The pulsejet was evaluated to be an excellent balance of cost and function: a simple design that performed well for minimal cost.[6] It would run on any grade of petroleum and the ignition shutter system was not intended to last beyond the V-1's normal operational flight life of one hour. Although it generated insufficient thrust for takeoff, the V-1's resonant jet could operate while stationary on the launch ramp."
Ok. I am starting to warm up to this. We could as POD in 1934 have:
a) The air ministry not care about the specifications (perhaps someone forward thinking champions its development based on future merits)
b) Have the original design put the engine above the fuselage
Even if we doubled the development time to 4 years, due to needs to refine the design we could have a production ready weapon in 1938. Perhaps with an additional POD of not messing up the Ju88 design, we have some spare aircraft production capability for this by 1940, end of May 40 major effort begins to produce large quantities of V1s as Germans plan for war with Britain after France falls.