BS”D
I am very familiar with the Georgian Jewish communities and I can say from the names of family members, first and last, that I am 90% sure the girl is not Jewish. The mother’s last name is most definitely not one associated with Jews, and assimilation in Gruziya was not to the point that we can even suspect that her mother’s mother could be Jewish as is the case with immigrants from other regions of FSU.
What is more, the Chabad shaliach to the Russian Jewish community in Toronto has a very strong connection with Georgia, namely that his mother AH was Gruzini, and as a very respected askan he would be the first one involved if the girl were Jewish.