I have never heard of the RAF pushing the Merlin beyond its normal operating limits with the exception of 12lb boost being used for longer than five minutes. That did happen a number of times.
As for the P40 in the BOB it was never going to happen being nine months too late for the battle. Even then it lacked climb, performance, protection, agility and altitude performance. The engine only produced 1040hp the 12lb boost Merlin was looking at 1,300hp a significant difference. A P40B could hold its own up to approx. 15,000ft but above that forget it and I think that only made service in mid 41..
I meant that the Battle of Britan (and prior action in France) forced the RAF and Rolls Royce to put more aggressive engine testing/experimentation in place, developing emergency ratings on engines years before the US did. Same engines ... all a matter of testing in the factory and in the field.
At normal 3000 RPM, the V-1710-33 was officially limited to only 37.2 mm Hg manifold pressure at 14,500 ft (that's +3.5 psi boost).
If overboosted to 46" Hg (just shy of +8 psi boost), it could pull 1,280 hp at 9,000 ft (same as the Merlin I's FTH for 1,300 hp )
That's without resorting to overspeeding to boost supercharger output. (3200 RPM makes the 8.8:1 gearing act like an effective 9.4:1 supercharger ratio would at 3000 RPM)
The 1st production P-40 was delivered at June 1940, the 1st Tomahawk I was delivered (still it is in the USA) at Sept 1940. Production-wise, they might qualify for the BoB, but obviously not when we count in the training and logistic needs.
The early P-40 will climb better than any subsequent P-40, being far lighter. With 120 US gals, the P-40 was at 6800 lbs and doing 357-360 mph tests.
Unless you count intial climb on some loadouts for later P-40s in WER power ... definitely the fastest climber on mil power.
Protection of 1940 BoB fighters was not equal with what fighters carried in 1941, let alone, say, in 1943.
I think the P-40B was fairly close to late model Spitfire Mk.Is maybe closer to Mk.IIs. P-40Cs got the totally redesigned fuel system using self-sealing cells rather than conventional tanks with sealing material applied to the exterior. (Cs also added armor glass to the windshields as standard)
The Spitfire is not the only aircraft that fought during the BoB. Early P-40 does have it's shortcomings, but it's performance was closer to the Spit and 109E, than to the Hurricanes and Bf 110.
Roll rate, overall control weights at high speeds, stall characteristics, and dive acceleration would be useful characteristics too. (plus lack of negative G problems) Then there's the higher fuel capacity and good fuel economy allowing a more flexible mission profile than the spit/hurri/109. (longer endurance, longer loiter time ... roaming patrols might be logistically difficult though -that range/endurance would have made it more interesting for the Germans as an escort or fighter-bomber compared to the 109)
The level speed the P-40 managed in spite of being heavier with lesser engine performance seems to point to it being a cleaner airframe than the Spitfire or 109 as well, and while that might not mean too much on the whole it does imply better dive acceleration than any of the BoB fighters. (the one place that higher weight becomes an asset)
Rear visibility is a bit better on the P-40 too, with the hollowed out and glazed fuselage section immeditately rear of the cockpit.
Power to weight ratio might have been worse than the Hurricane, but the P-40B's climb rate figures seeem to compare favorably to the Hurricane Mk.I, though sustained turn-rate was worse. Quite possibly a better interceptor than the Hurricane, though not the Spitfire. (ironically, the 109E's performance and armament best fit the bomber interception role of the lot)
Nevertheless its interesting to note that many of the engineering decisions made by RJ Mitchel and J Smith over ruled the possibility of using the full wing for fuel storage. For instance Mitchell did not like the idea of synchronised cowling machine guns or a motor gun because it inevitably 'thickened' the fuselage. Had weaponry been mounted on the cowling, between the V block or in the wing roots (as in the Fw 190, He 112 and several Soviet aircraft) the wings might have been left clear. There were no doubt other issues such as lack of a suitable guns, the inability of Merlin engine propeller combination to take this without re-engineering. The US 0.5" Browning with hydraulic hydrostatic interrupter gear in the wing roots seems the easiest to me. Likewise the rejection of 'gondola' hispano weapons on the Spitfire by J Smith.
It seems neither man was motivated by the Air Ministry to increase the range of their aircraft.
The .50 browning or (if they'd taken an interest) .50 vickers (might have had better synronized RoF too and lighter and more compact -2 wing root guns plus 2 cowl guns would have been decent even with the lower powered vickers rounds).
Mounting hispanos in underwing pods was suggested too. Use fuel tanks in the wings and a pair of 20 mm underwing and it might have been useful. (depends just how much drag the installation added)
Tony Willaims has a couple articles pointing out the realistic potential for the hispano to be made synronization capable too. I'm not sure about engine-mounting it though, that'd depend on the Merlin's design. (the V-1710's supercharger arrangement didn't allow this ... neither did the propeller hub positioning used on the C series V-1710s) It might have been a possibility though. A mix of HMGs and nose cannon would have been pretty useful too ... and allowed WORKING cannon-armmed single engine fighters during the BoB. (1 20 mm plus 2 or 4 HMGs is a pretty useful fighter armament)
Somewhat more puzzling is the lack of a long range escort specification by the RAF unless once considers the Typhoon and its teething problems.
The Typhoon's range hardly fit the requirements for that role either, even without teething troubles.
My understanding is that the Tomahawk IIA (P-40B) carried 131 Imp Gallons (160 US Gallons) but that later versions dropped down to 120 Imp gallons) This is still well above the Spitfire III/V's 85 gallons and similar for P-39.
The P-40C carried tanks of 134 US gal (111.7 Imp), but the P-40E expanded that to 157 gallons (130.7 Imp). The P-40N reduced that to 122 US gallons. (on a side note, if the same proportions are applied for the 160 gal vs 134, the P-40E's configuration might have allowed some 187 US -156 imp- gallons with P-40B style tanks)