We did exactly that last summer, and Georgia was a huge hit with both adults and kids. What helped us a lot was using  https://premiumgeorgia.com/ because they put together a custom plan. We had private transfers, which saved the stress of long drives, and stayed in boutique hotels with pools (great for the kids). The highlight was a cooking workshop where the kids learned to make khachapuri with a local chef – they still talk about it. The balance of culture, food, nature, and comfort was spot on. From London, the flight was shorter than I expected, and once we got there everything was seamless. It felt like an adventure, but still luxury.