Well, I did write that the RN was allowed to spec out their fighter and the Admiralty took control of the FAA in 1935 instead of 1939. I do appreciate all the input and discussion, the different perspectives make interesting reading. I always wondered what else Mitchell may have come up with if he had lived longer, he was such a good aircraft designer.
I agree he was a great designer. And it would be very cool to see what he contributed to thereafter. I just don't think it would have been a naval fighter. AIUI by the time he was forced to stop working he was involved in the Type 317 4-engine bomber project. Had he continued working he likely would have stayed with that project. Though it is likely still killed by the bombing raid destroying the prototypes.
Mitchell's death did make the AM more leery of the 317. If he had lived maybe they would have been able to get further along, faster and been chosen for production before the bombing. It seems like that might take an unreasonably fast development though.
During the early war, it seems likely that most of Supermarine's resources will be focused on expanding Spitfire production. Mitchell may look at follow on designs to the Spitfire sooner than Joe Smith did, which could lead to a Spiteful analog soon enough to make a difference before jet engines take over. Maybe the MAP even lets them make a Seafang analog for the RN.
If you are looking for an early single seat fighter for the RN, supposedly Fairey (an approved naval aircraft manufacturer) proposed a fighter in 1934 (the same timeframe as the Spitfire and Hurricane started life) with their own P12 or P16 engines (supposedly the P16 offered more power than the Merlin for less weight). The design was offered with an Oerlikon cannon firing through the spinner and two engine mounted machine guns. I don't know much more about it than that, and Fairey's abilities at design outpaced their production facilities, but a Fairey design that doesn't immediately compete with the RAF's resources for Engines or designer time seems like it would have a better chance of getting into the air.