I have the WS1200-Airbridge/Wifi version. Same sensor module as yours. There is a "low battery" indicator on your console that will be to the lower left of the circular wind direction display.
I had a different experience with battery charging. I have only had my system for a month, and we had a lot of overcast days. Then we had 4 solid days of rain with no sun, and my sensor unit died out with "no power" on the 4th day in the middle of a storm. I had to bring it inside and begin recharging the unit with interior lights. I then decided to temporarily swap in some non-rechargable batteries until the storms cleared up.
So if your unit is fairly new, and the recharagable AA batteries that came with the unit have been sitting "idle" for some time on-the-shelf, they may not have had sufficient time to soak up a full sustained charge if you get do not get a lot of good-solar days in the beginning.
As Arktikos mentioned, getting some spare rechargable batteries and a charger is a good fallback. I've also determined that the sensor can be powered with an appropriate AC adapter and *long* power cord, which I'm thinking of implementing.
You may also find that if your system goes for quite a few days without decent sunlight you might start getting missing sensor updates (that happen normally every 16 seconds). I've been able to watch the signal strength of my sensor's RF transmitter (which the "AirBridge/ WeatherBridge" unit provides), and that signal strength does decline quite a bit as the batteries weaken if they do not get sufficient recharging sunlight over several days.