OK, checked with my friend in the Georgian consulate again and he sent me Georgian legislation in MS Word, in Georgian language, of course. I don't know how much Georgian you speak, so I'm translating:
Law on the Legal Status of Foreigners in Georgia
Chapter II, Article 3 (Principles), Provision L
"Persons having permanent residence of a foreign state, who do not hold citizenship of that state, are subject to the same legal provisions that are provided for the citizens of such state".
And because the US citizens don't need a visa to visit Georgia, the permanent residents don't need it either. Besides, he told me that the consulate received a memo couple of months ago, which specifically outlines procedures for the Green Card holders and it's exactly what I outlined above. All you need is your Green Card and travel document.
Hope this helps. Have a nice trip.