We stayed at Castillo di Verrazzano and had a wonderful time, but I don't think it would be a good fit for you. There's not a pool there. We used it just as a home base. Also, it has a pretty steep drive from the main road. Bicycling up and down would be a challenge.
Speaking of bicycling, Chianti is pretty hilly. You'll have to choose carefully where you stay. Even then it will be hard to visit many of the fun hill towns by bike.
Since you'll be there in July, be sure to check minimum length of stay. Many places have a one week minimum in high season.
You really need a car to explore Chiant/Tuscany. Train service is limited and you'll need to be able to go to a town to catch it. Verrazzano is just outside Greve and there is bus service. We took a bus into Florence, which was fun since it is a commuter bus and we were the only non-locals on it.
Finally, to each their own, but we find the further up the luxury ladder you go, the more isolated experience you have. We make a point to stay at smaller accommodations that are clean but modest. At Verrazzano we made fast friends with the staff. Marco and Isabella looked after us. At breakfast, Marco would practice his English with us and he taught us Italian in exchange. When we were in Scotland, we stayed at a small B&B. We helped our host Peter cook fresh crab salad on his Aga stove. In London we stayed at the Westminster Hilton. It was posh, but very boring.
