That's a lot of battery water.
I do not believe many solar stills are well though out.
To condense water from air is easy. A glass of ice water does it because it is cooler than the air.
Most solar stills are at nearly the same temperature over the entire unit. There is not much more reason to condense than there was to evaporate.
It would work better if the unit had a cool area. That should not be difficult.
The area where the condensation is collected can be shaded.
Possibly some kind of simple heat sink, even extend it into the cool area. Back to back CPU heat sinks with one inside and one outside?
Possibly evaporative cooling of the cool area. Wrap the area in denim, keep it wet with wicking action or a slow drip, maybe supplied from one of those automatic dog bowl fillers that use an upside-down 2 liter soda bottle reservoir?
Simple idea, just for the concept. 4" PVC in an "H". Cross piece slopped down toward the cool side. 1/4" plywood between the vertical sections to provide shade. Paint the hot side black, maybe add a horizontal trough in the vertical piece at the lower edge of the horizontal piece. Maybe add some cooling to the cool side.
Evaporates from the hot side, condenses in the cool side.
I expect the solar still in Zap's link would work better if there was no insulation over the short side, and maybe it could use some kind of heat sink on the end or lowest 3" of glass (wet denim?).
G-