Is it permissible to calculate transportation allowance from one's original residence in the countryside, and not from the temporary residence in the city, even though the direct manager has authorized it?

If the authorized person in the company grants you permission to take the transportation allowance based on the mentioned calculation, then there is no harm in it for you. However, if the matter will require writing invoices that do not match reality, then this is not permissible due to the lie involved. There is no need to lie by writing unreal invoices if the person is authorized to give permission.