I was thinking of this thread again (the tug-of-war definition) and thought of another interesting thing to calculate: What if I swap Pluto and Charon in the equation? Would it indicate that Pluto is also more influenced by Charon than the sun at times?
The answer is yes, meaning Pluto does not have a concave orbit either. The tug-of-war value focusing on Pluto as the primary body is 337.3 at perihelion, but focusing on Charon as the primary body it is 39.3. Since both are greater than 1, this means Pluto and Charon cause each other to have an occasionally convex orbit.
So perhaps this definition would still find Pluto/Charon to be a binary planet, but in a different way than the Earth/Moon are. Compare the numbers above with Earth/Moon: Earth-centric has a value of 0.45, Moon-centric has a value of 0.006. Both being less than 1, Earth and Moon always have concave orbits around the sun.