Would Byzantium also have the option to give acces and for instance when Egypt is conquered by the Italians, would they be forced to join Axis or face defeat? An outright alliance for Germany would gain nothing for both sides yet.
Another thing that I thought about was the Syria part. Would it be just how Japan would occupy French Indochina. So when France falls, Byzantium attacks the remaining part of Turkey (very easy though, one needs only to conquer Ankara and annex

). So when Byzantium attacks Turkey and wins and France falls, Syria is kinda left under a pro-Vichy government with little help. The same goes for Indochina and gets occupied by Japan (in this case it is made puppet).
What my proposal would be is:
1. outright annexation demand (which would be 85-90% chance be refused by Syria) - when refused, Byzantion gains option to attack -> Allies react to this with a chance that UK says war!
2. the second option would be a forced puppet (this would be more acceptable to Syria say 10-15% chance refuse) and a year or two later they are annexed to the Empire (with potential revolt or war but small chance)
3. Do nothing (remain isolationist) very low chance...
last but not least. There should be an event that replaces the PM of the Empire Konstantinos Demertzis (dies in office) with Ioannes Metaxas (very pro monarchist / imperialist)
Tim