Hi, the different time zones of this globe can be a real pain.
notjarvis brought up the most important item, your Line manager is fine with this, i.e. the decision is allredy taken and the director just need to sign off and not do any thinking himself, hopefully.
BUT, why does you manager need the director to sign this of, there could be a mine (prox1) burid here?
Here lies the solution to your problem most likely. If you can probe you line manager more about this you will be able to prepare better for victory. It could be many reasons, some better and some worse.
Some examples:
a) this has been brought up before by the line manager (without asking the director) and the director dissaproved of it then, manager want's to check if status has changed but is to much a coward to go and hask him/her self
b) line manager does not have authority to re-schedule work hours, this is simply protocol
c) line manager is a coward and don't want to take "big" decisions him/her self, director probably knows this, this could go either way
d) this is actually against policy, even though your manager don't dissapprove or understand why, you'll need to change policy at your work place
Lying is usually a bad idea. Not telling the whole truth is no the same thing and can be used to your advantage.
If you can find an activity that is held monday mornings and which indirectly benefits either your health or your ability to perform your work, your director will most likely be more inclined to approve the change of the schedule if he can (see policy issue above). Fist you play Allegiance, then you go to your activity (most mondays anyway) and then you go to work.
Going to the gym to keep fit is usually a good thing, the trick would be to find a reason to why you can't do it in the afternoon or early morning like most people. Perhaps a class in management training, gardening

are better alternatices, IF any are held monday mornings.
Taking language lessons with a private tutor could work if you for instance have a recognised minority in your community, but could be expensive and is extra work.
I hope it is just a protocol thing, then stating that you just want to train physics together with your weekend rugby team buddies (or similar) on Monday mornings (even though you then afterwards suddenly quit this imaginary activity) would be reason enough.
Good luck!