Would refrigeration lengthen shelf life for this water sweetener?
The most likely answer here is that the flavor compounds either degrade (oxidize) or evaporate over time in an opened container. Refrigeration is unlikely to change this a lot as the rate of evaporation is dependent on the diffusion gradient A large gradient = faster diffusion, which would be determined by the head-space in the opened bottle.

Oxidation processes are dependent on the amount of substance to be oxidized and the amount of available oxygen, though there also things like free-radicals which can oxidize one molecule and then move on to the next. Temperature might slow this a little, but probably not so much that you would notice greatly.