I had a look at my rental car agreement for October, i.e. returning the car on 30th and returning to Greece on 31st. It's an early morning flight so there is a good chance of getting back into Greece before midnight when they close and padlock the door even if there is some delay. It does contain the following condition:
1.2 Licence must be in Roman letters, Latin alphabet or accompanied by an International Driving Permit (IDP). IDP’s must be accompanied by the national drivers licence.
Your hire company may have a similar condition and may be treating it too literally? While it is true that the Greek licence does contain Greek letters, e.g. ΑΔΕΙΑ ΟΔΗΓΗΣΗΣ ΕΛΛΗΝΙΚΗ ΔΜΟΚΡΑΤΙΑ, it is in exactly the same format as a UK licence. Looking at the symbols on the back it is clearly a driving licence and GR superimposed on the EU flag in the top left corner is the recognised symbol for Greece. All the information is next to numbers that are the same as the UK numbers as far as I am aware. Your name should be in the Latin alphabet. Virtually everything else is either numbers, e.g. dates, or symbols. Mine has been accepted multiple times by several different hire car companies over a 14 year period with no IDP.