The model shape doesn't get changed when doing the manual rereferencing other than getting stretched a bit in X and Y. The elevation offset that gets applied is just an offset and tries to minimize the error between the values you supply in the GCPs and how we can offset and rotate the model to fit it. They will not be exact. We optimize for accuracy in latitude and longitude first and elevation second.
But, the volume calculations will still work. They are a relative measurement. So assuming the elevations are not WAY off or the tilt is really bad the accuracy of the volumes shouldn't change much. You can compare it yourself by making a measurement on your original and on the realigned version and comparing them.