It's been a while since I got some stick time in a K-4 but here's what I'd do against an F4U (or any other turner). I start by making B&Z passes, but I don't actually intend to get a gun solution. A high energy turner is far too difficult to hit. I only B&Z him to make him expend energy in evading, and I make sure I break off in time to be out of guns range when he turns back into me. I will repeat these fake attacks until he's on the deck, slow and helpless before making the kill pass.
This assumes that the K4 enters the fight with an advantage. The K4 would be in just as much trouble if the positions were reversed - with consideration given to the K4's ability to regain E faster than the non-4 Hog's.
Using the K4 as a "B&Z" plane is a faulty use, IMO. Just the same, considering the F4U a "turner" is equally faulty. The Corsair can outdive and out zoom the K4 with relative ease due to weight and wing design.
The K4 is a horrible diver and its advantage in ROC vs the F4U - in actual use as opposed to on paper - is only at lower speeds due to the aforementioned zoom ability of the Corsair, which is not measured.
In other words, both aircraft being at 400 knots, I wouldnt try climbing away from an F4U because it would follow just fine. However, at 200 knots, a spiral climb is a fine play and - even then - you run the risk of the F4U gaining a flaps-fueled gun solution, anyway, as Agent pointed out.
Regardless, the two big advantages that the K4 has over much of the plane set are both found at low speeds; the ability to take a low-speed fight into the vertical and the ability to accelerate very quickly.
At high speeds, the K4's only advantage is the actual indicated/true air speed, itself. The airframe does not handle high speed well and the weapons might as well be on 'Safe' because youre not hiting anything thats not AFK with a tater at 400MPH.
With that in mind, against the F4U, you simply avoid his strengths and exploit his weaknesses. That means forcing the fight low and slow, and then taking it vertical. While the Corsair may hang on with flaps, the K4's ability to bring the nose back down in this situation, with engine torque (or lack thereof on the opposite direction), is the key to a quick victory.
If the fight is at altitude and the K4 can not convert to a guns solution quickly, the Corsair is very capable of holding E in the opposite manner precribed above. It will outdive the K4 and, as previously mentioned, will then proceed to outzoom the K4 (if the K4 doesnt compress and lawn dart) - reversing any advantage previously held.
Point is - I wouldnt be so quick to discount the Corsair as unable to contend with a K4. Im pretty far from a top-tier 109 driver - hell I suck compared to most - but there are more than a few F4U sticks that give me fits. Dogg and Vudak come to mind, for example.