To try and explain the apparent discrepancy between map and drawing, one should go and find out the process of creating the map.
The aerial images are taken individually from a great height. But the images are not orthographic and there is distortion increasing from the centre of the image. The map is made up from many images, which are put together and the surpluses discarded. So the distortion is not constant throughout the mapping.
I expect the positional accuracy at maximum distortion is a closely guarded commercial secret, but it would not surprise me if it were to be plus or minus several metres. So even if you could match one part of the mapping, then 500 meters away, the mapping would not match. There is nothing you can do except accept the imperfect world we live in.
When I said 'nudge', I did not realise there was a command to do this, but I was thinking a moving the map a little bit. If you cannot select the map, there is nothing I can advise you to do.
I would take a screen grab of the mapping so that you can manipulate it. Someone else may know different.