I'd also call the Georgian Embassy in your country, not to be a smarta@@ at customs, but to enable you to find out more info to comply with their rules. As Nigel said, countries of the former SU can be difficult, but no more so than many others.
Not to upset anyone - but I have had more trouble getting through US Officials than many other places ... in fact I was arrested one time when flew into Washington ...
They said my ESTA wasn't valid with the Passport I had ... (Pre Biometric) ... so refused me entry ... then said I would be deported back to origin !
I asked if I could sort a visa ...
Reply : No because Visas are not issued on US soil ...
I replied : But Maam - I am not in USA yet - I am in no-mans land ... behind me is technically where I came from - over that line there is USA ... so can I have a word with Supervisor - see if any way round this ?
Supervisor arrived ... took my passport and myself to detention office ...
I was fined $60 for entry without valid Visa ... Airline was fined $3000 for flying me there ... (note that Dept of HOmeland Security was at departure airport gate checked all passports including mine and no mention of problem !).
Supervisor called for my bags ... then told me I had 10 days entry to USA ...
Passport was stamped "Paroled" and entry to US Immigration Computer !
I only needed about 5 days anyway - over-seeing delivery of 100,000 tonnes of Fuel to Gulf of Mexico ports ...
I know - nothing to do with ბათუმი
Nigel