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The bigger issue is weight. Start by considering that the original Allison V-1710 engine weighed about 1400 pounds (29 inches wide generating up to 1700 horsepower for take-off), while the radials weigh 2,000 and 2,300 pounds respectively. The extra weight would require a larger wing to keep landing speeds short enough for existing runways.
Wright R-2600 Cyclone 14(Twin Cyclone) engine produced 1700 horsepower for take-off, but weighed 2,000 pounds and was 55" diameter = not much advantage over the original Allison engine.
OTOH Pratt's R2800 engine produced substantially more power: 2100 horsepower (2400 pounds, 53" diameter). It had (almost the same as the other radial) but substantially more horsepower than the original Allison. The disadvantage is that R2800 weighs a thousand pounds more than the Allison and drinks a lot more gasoline .... meaning a much larger airframe. ..... larger than Kingcobra (1800 horsepower 8800 pound gross weight).
Let's look at engines of mid-1941.
The original V-1710-35, on the P-39s of the era, was good for 1150 HP for take off and 1150 HP at 11200-12000 ft (different tables disagree); 1375 lbs dry + weight of liquid cooling system (was ~325 lbs on the P-39) = 1700 lbs
The R-2600-9, used on early B-25s, was making 1700 HP for take off, 1450 HP at 11800 ft; 1980 lbs.
R-2800-5, used on early B-26s, was making 1850 HP for take off, 1500 HP at 15000 ft. All figures are 5 minute ratings (military power at altitude); 2270 lbs.
Weight difference does not seem excessive - around 200 to 600 lbs. Power figures are far better for the radials, especially for the R-2800. V-1710 will gain more power at lower altitudes once war sets in, in a more or less official manner, however the R-2800 will get an upgrade in winter of 1941/42. Having no wing armament means more fuel can be installed there instead.
Consider the difference in weight between Grumman's Wildcat (1200 horsepower and 8800 pound gross weight) and Hellcat (2200 horsepower and 15,400 pound gross weight).
Early Bearcat - 9670 lbs clean, max fuel & ammo, 2100 HP mil power.
And this uber-Cobra would still need 20mm or 37mm cannon(s) to intercept any Axis plane built after 1940. It would still be possible to install a 37mm cannon. Just chose the cannon with the best ballistics for tank-puncturing or bomber-busting, then install it in the nose the same as the original Aircobra.
As long as you installed a drive-train similar to M-18 Hellcat (tank destroyer AFV) you could keep the drive shaft on the cockpit floor and still leave plenty of room for guns in the nose.
I'd went for armament nominaly similar to the Piaggio P.119 - one 20mm cannon whrough the prop, 4 .50s inthe nose.