Logistically, it would have been difficult. They might have been able to stage from the Italian-held island of Rhodes, though I don't know what kind of air facilities the Italians had there. Having an airborne division to use would require either not going after Crete or having a much better outcome there--probably a better outcome than they were likely to get unless they trapped a larger part of the British army in Greece.
Assuming they still had the division to send, a lot would depend on how much collaboration they were able to coerce from Vichy-held Syria. If they could gain access to airbases and ports in Syria and use aircraft based there and in Rhodes or Crete to dominate the nearby sealanes they might be able to sort of support the force, though at a huge cost to their ability to supply North Africa.
The obvious British response would have been to reinforce their forces in Cyprus, which would have probably played a role much like Malta did, assuming that the Brits could hold the island.
As a best-case scenario, this over-stretches the Brits in the short-term, but then becomes an open sore for the Axis logistically and saddles them with some seriously incompetent allies in Iraq.
The big unknown here would be how the Turks would react. This would take the fighting into the Turkish backyard, into areas they had held prior to World War I and where they still had territorial ambitions, especially in parts of Syria and northern Iraq. If the Germans somehow managed to persuade or coerce the Turks into coming into the war on the Axis side or allowing German supplies to flow through Turkey, that would make holding Iraq a bit more feasible, though it would still be a drain on the Axis.