Yeah, yikes! I forgot about the Goshawk. That definitely does illustrate the problem. Which in aggregate, does mean you needed not just a slimmer fuselage, but a redesigned wing with somewhere to put the main landing gear other than in the fuselage. Which means basically a whole new plane. A Hellcat, in other words.
The Corsair has a thinner fuselage (max width) than the Wildcat too I'm pretty sure, in spite of having a bigger engine with a wider diameter. So it's definitely doable.