I used the service of the Armenian car rental company: http://caravan.am/
I paid them to bring a Lada Niva to Tbilisi. From where I would drive in Georgia and Armenia. At the end I brought the car back to their office in Yerevan.
They provided me with the required car documents to cross the border myself from Georgia to Armenia.
From my experience I will like to advise you NANIKO car rental in Armenia. You can rent a car in Yerevan and leave it in Tbilisi.
Also as I compared early (5 month ago) NANIKO was cheapest, so highly recommended.
Hertz does, although I haven’t tried myself. You can check Hertz’s Armenian website (in English), where you’ll also get the price and be able to book this exact service.
Hertz’s Terms and Conditions even has a special section on renting in One Direction specifically for Georgia.
Credit:stackoverflow.com‘