Firstly, this depends on the angle between the orbit and the reference plane (horizon). For instance, a perpendicular error in an orbit perpendicular to the horizon will result in a almost purely azimuthal error, unless close to zenith. In contrast, an orbit with a low maximal elevation will have almost only elevation errors.
Secondly, it is more convenient to use an angle instead of a distance to represent the error as that makes the calculation independent of altitude.
Looking at the problem now, we can see that using the initial orbit as the reference plane, the error is just adding elevation. The problem is now reduced to a simple transform between two spherical coordinate systems. To keep this transform as simple as possible you should use the ascending node of the orbital plane as your reference direction for azimuth in both systems. Getting a final azimuth value can then be done as a final rotational transfer along the polar axis, meaning you are back to your initial system by subtracting the longitude of the ascending node.