I have no information myself, but I'd be willing to bet a million dollars there were plenty of officers/officials that were very much against the turret fighter.
That's a little difficult to justify; the power operated turret was considered by the heads of the RAF as the last word in accurate weapon aiming and defence pre-WW2, hence the Defiant and Roc (although BP didn't want to build the Roc, having submitted their own naval design to have it quashed in favour of the decidedly inferior Blackburn machine. BP's head, J.D.North was not very happy at all about the decision, apparently).
Regarding how highly favoured turrets were, Ludlow-Hewitt C-in-C Bomber Command, Tedder and Sholto Douglas wanted the DH.98 Mosquito to be built with a power operated tail turret and it was in this form that the go ahead to procure it was made - it was only after Liptrot had agreed to a night fighter variant, to which a spec was written specifically for the Mossie, that an order came through for production - GdeH was pressing on with the unarmed 'concept demonstrator' despite the turret armed aircraft; Freeman stepped in in favour of this. So, tall and short of it was that the power operated turret as a defensive weapon was an advanced concept, so high up in official circles, it was the way to go, and the concept of a turret armed fighter fit in with what was considered advanced at the time- the primary bomber contracts, B.12/36 and P.13/36 specified turrets - that is, until the shooting started.
Perhaps bringing BP into the mix might make more sense if the firm was allowed to build its own naval fighters; there was a variant of the Defiant proposed as a carrier fighter, the P.85 Sea Defiant
without a turret and fitted with forward firing guns. Alec Brew in his book The Defiant File argues that had the P.85 been built instead of the Roc, the FAA would have had an effective single seat fighter that would have meant there was no need for the Fulmar, Sea Gladiator
or the Sea Hurricane.
I see what you are trying to do Tomo; in an ideal world an earlier Sea Hurricane could have been a benefit as the war progressed, certainly in the Mediterranean. The big problem the RN was faced with was a shortage of aircraft of any type - there simply weren't enough Fulmars, Gladiators or Skuas to go round and I doubt that would have changed had the Sea Hurricane entered service earlier. One issue that might have had to have changed was that the Sea Hurricane didn't have a folding wing. This would have been a benefit in the small spaces of British carriers.