Praise be to Allah.
If the matter is as you described, then you should go to the responsible department and have the medical exam, and thus the money will become permissible, doubt will be dispelled and you will be free from wrongdoing.
It seems that their stipulation of a medical exam is that it should be done by a department of the ministry.
If you are abroad because of studies that prevent you from attending and having an exam, then we hope that there will not be any blame on you if you take the money, so long as the description applies to you.
And Allaah knows best.