I think you mean "both would not be considered inertial in GR". An inertial frame of reference suggests the existence of a homogeneous, isotropic region of spac

I think you mean "both would not be considered inertial in GR". An inertial frame of reference suggests the existence of a homogeneous, isotropic region of space; in general relativity, gravitation shapes space and, since gravitation has infinite range, such a region does not exist. The closest relativistic thing would be a reference frame free-falling inside a gravitational field: it is locally inertial. So yes, the Sun is more inertial than Earth, but neither are a real inertial reference frame — because such a thing cannot exist. In Newtonian perspective, they can both be considered inertial — but you will also need to pick three stars far enough so they seem immobile.