The cone of copositive matrices is convex, but it is not a symmetric cone. So the common notion of "geometric means" on this cone will look different from geometric means on symmetric spaces. See for example, Section 2 of this recent paper for some discussion of geometric means on symmetric spaces.

Even if you were able to define a Cartan mean on the set of copositive matrices, it'll probably be uncomputable, and therefore not so useful (primarily because testing membership in this cone is hard).

A related, interesting direction worth considering might be (if you are willing to tolerate NP-Hard things) to instead consider the cone of double nonnegative matrices; this is (as you are I guess well aware) a subset of the CP matrices, while still being much more computationally tractable.

The cone of copositive matrices is convex, but it is not a symmetric cone. So the common notion of "geometric means" on this cone will look different from geometric means on symmetric spaces. See for example, Section 2 of this recent paper for some discussion of geometric means on symmetric spaces.

Even if you were able to define a Cartan mean on the set of copositive matrices, it'll probably be uncomputable, and therefore not so useful (primarily because testing membership in this cone is hard).

A related, interesting direction worth considering might be to instead consider the cone of double nonnegative matrices; this is (as you are I guess well aware) a subset of the CP matrices, while still being computationally tractable.