To date, we cannot even prove by observation the Oort cloud even exists.
And there is good reason to believe that if it does exist, it is an Oort Mist rather than a cloud. Many believe if anything exists that far out, it is being replenished by the Hills Cloud*, and is not a static reservoir. The Hills Cloud is, loosely, a distant but separate "extension" of the KB in that it largely resides along the ecliptic, and is much closer to the sun than any hypothetical Oort Cloud. The Hills Cloud is projected to have trillions of icy, comet-type bodies.
The Oort cloud is much more contentious, particularly notions that it extends out to 3 ly. If there really is/was a spherical Oort Cloud and it was at the projected extreme distances from the sun, it does not stand to reason that much of it would remain after 4.5 billion years of traveling through the galaxy, with gravitational perturbations from within and without the solar system.
Picking out any interstellar objects from the "native" bodies would not be easy. And their numbers are not expected to be very high. Funding for such a mission, to a hypothetical object, would be difficult to obtain. Perhaps high resolution telescopes of the future could resolve any issues of identity or origin.
And many (more than likely) "native" objects orbit outside the ecliptic. These are found in the Scattered Disc**, or are defined as "Detached Objects"***.