If Pluto is a planet, it seems reasonable that something that's bigger than Pluto, and further away than Pluto, should be called a planet, too.

The answer is that we have surveyed roughly half of the sky and found one object bigger than Pluto and two that are almost the size of Pluto. As we finish the other half of the sky we will almost certainly find a few more of around the same size and maybe one or two even larger.

There is nothing particularly special about that location of the centre of mass. If you were to find yourself at the precise spot that is the centre of mass of the earth-moon system, the only thing unusual that you would notice is that there would be one thousand miles of rock on top of your head.
Pluto is only about twice the size of Charon, so if you put Pluto and Charon on the cosmic seesaw you would find that the balance point is a little bit outside Pluto, rather than inside it. Again, there is nothing particularly special going on there. If you were to find yourself at that precise spot, you would only notice that you were very, very cold and could no longer breathe.

All three objects are nearly Pluto-sized or larger, and all are in elliptical orbits tilted out of the plane of the solar system,

I refuse to accept Pluto's resignation as a planet.

The wrong way always seems the more reasonable