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changed the question somewhat
Cornelis
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Why not interpret the Kármán line as a realistic, curved line?

$$\frac{\rho v^2 S C_L}{2} + \frac{m v^2}{r} = \frac{GM_Em}{r^2}$$

$$v^2 = \frac{GM_E}{r (1 + \frac{\rho S C_L r}{2m})}$$

Why not interpret a Kármán line that takes the centrifugal force into account ?

According to Wikipedia's article about the Kármán line:

The Kármán line is the altitude where the speed necessary to aerodynamically support the airplane's full weight equals orbital velocity ( assuming wing loading of a typical airplane )………………………………………………………………………………………………………………...The Kármán line is therefore the highest altitude at which orbital speed provides sufficient aerodynamic lift to fly in a straight line that doesn't follow the curvature of the Earth's surface.

Why not just follow the curvature and take the centrifugal force into account ?

Suppose the aerodynamic lift force $$F_L$$ of the plane would equal that centrifugal force $$F_C$$ ?

Looking at the expressions of the question:

"What would a Kármán plane look like, a bird, or a plane ?" we can see what would change !

If $$F_C = F_L$$, then ( with $$F_G$$ being the gravitational force )$$(1) F_L = \frac{1}{2}F_G$$

Substituting expression (1) will give a new orbital velocity that is the square root of the former one, so:

when $$F_C = F_L$$ then the orbital velocity will become about 30 % lower.