As this is a popular topic, I decided to look for current research. As I expected, there are very few studies about gout and carbonated water. However, I was surprised by the following (from Drinking mineral waters: biochemical effects and health implications – the state-of-the-art in the International Journal of Environmental health):
it is suggested to drink these waters as they gush from the spring to avoid the element decay and thus the therapeutic properties decrease. […] The therapeutic property of radioactive waters seems related to the energy released by the radioactive elements that yield excitation and ionisation properties. The therapeutic uses of radioactive waters concerns the treatment of osteo-articular diseases, gout and of diuresis stimulation (Gans, 1985; Bartoli et al., 1989; Amrani and Cherouati, 1999).