You mention a couple of times the curvature being an invariant. I wonder if you're confusing this with being dimensionless. I guess that you're talking about the Ricci scalar, which is a relativistic invariant, but it is not dimensionless; IIRC it has units of inverse distance squared. On distance scales much smaller than scale implied by the curvature, the curvature will be negligible and will not be detectable. For instance, a curvature of 1/(1000 km) will be easy to see on scales of 1000 km, but hard to see on scales of 1 m. That's why you don't see the curvature of the Earth's surface when you stand on the ground and look at things a few meters away.
When the equivalence principle talks about "local" measurements, it means measurements taking place over a small-enough region of spacetime that the curvature is negligible. You can always restrict yourself to a small enough region (formally, an infinitesimal region) so that spacetime looks flat, no matter how strong the curvature or how sensitive your instruments. So, you can't discover the curvature by doing "local" experiments in the sense of the principle.
The content of the equivalence principle is then that freely falling in a gravitational field is (locally) indistinguishable from freely floating in space. In other words, the gravitational field can be made to vanish within an infinitesimal region by finding the appropriate reference frame - a freely falling one. There is no "leftover" part of the gravitational field that cannot be removed by freely falling.
Note that this is not generally the case for other fields - for instance, electromagnetic fields cannot be made to vanish by choosing a particular reference frame. A pure electric field in one frame may appear as a mixture of electric and magnetic fields in another frame, but if there is a nonzero field at a point, there will not be any frame in which both electric and magnetic fields vanish entirely (even locally). So, gravitation is quite special in this regard.