The weather in Batumi (where I lived for two years) is usually warm in late September (though, of course, there are no guarantees...) Beach-season really only runs through mid-September, though, so some of the summertime cafes and discos on the shore may already be shut for the season. However, there are still plenty of restaurants and cafes that are open year-round.
I'll add, however, that I very seldom ventured in the water in Batumi, as it was none too clean, and the rocky beach isn't very nice anyway. If you want to swim, better to hop a marshrutka (mini-bus) down to Sarpi or Gonio where the water quality is much better.