I never said it was impossible, I merely indicated that there were some considerations. Your proposal certainly has merit! First we would need to establish a horizontal plane on which to mount the sine bar. Because we don't want to inject error when we add the mass of the sine bar and AH mirror, we'd probably want to use a small surface plate.
In setting the optical path, which now demands that we adjust the AH mirror away from parallel to the horizon, we must understand the contribution of each of the two new angles. The in plane tilt angle (pitch from the viewpoint of the observer) and out of plane angle (roll from the viewpoint of the observer). In the nominal AH arrangement, both of those angles are 0°. Not so once we inject a tilt.
The actual in-plane tilt angle of the AH just has to be known accurately. It doesn't have to be set to some arbitrary angle, any angle will do provided the direct and reflected objects are visible. It certainly has to be known to the resolution of your altitude measurement or 0.1' of arc. Otherwise, the uncertainty of the in plane tilt contributes directly to the uncertainty of the measurement.
In terms of the azimuth, again, it doesn't matter if it's due north or due south or any other direction. What is important is that the in- plane tilt is within the vertical observation plane. That is, a perpendicular to the AH mirror, is part of the plane formed by the eye, the object and the point of incidence on the mirror. Thus we will not inject any error by out of plane tilt. Again, it is not the absolute direction, merely that the direction of the azimuth selected does not inject error.
The azimuth selected will only be good for a relatively short period of time. More precisely, it will have 0°of out of plane tilt for only one instant of time. But as a practical matter, the error injected by the out of plane tilt is by the cosine, thus we would have several minutes of observations before the error contributes a significant value.
I do hope I've communicated this well enough to be understood