Okay, so I can answer my own question.
It seems that it's difficult to get any information about customs even when you're in the country.
The lowdown is this: like @Johnnyjanko said, your vehicle has 90 days (or 60 days apparently to some locals) after crossing the border by land, which will likely be different to your personal visa. Any duration outside of this timeframe will be fined at 50 GEL (~$20) per day up to a maximum of 1000 GEL on your vehicle.
When leaving by air (which I have done) nobody asks you any questions at the passport control, although I had a letter from the person where the bike is kept to confirm there is someone responsible during my absence.
That's it, it seems you don't need any permission. But if you go beyond the allowed duration, be prepared to pay a fine when leaving the country.