High temperature battery chemistries
Shade is the first thing to try, e.g. from the solar panel. If the solar panel really is re-radiating (as you say in the comments), a sheet of space blanket on the back of it would takle care of that.  

A shiny or even pale device will get a lots less hot (unshaded) than a black one.  Of course if the air temperature exceeds your cutoff you'll still have a problem, though this might be solved by thermal mass (a slab of aluminium) if the nights are cold enough to lose heat.

It's a pity your thickness requirement is so low - 6V lead acids are available at 34mm thick (so are the 2V individual cells).  And of course lead acids can run quite hot under car bonnets.

You could provide air-cooling without unsealing the case if you press the battery against a heatsink set into the case wall, with a fan on the outside.  The fan would be exposed to dust but nothing else would, and IP55 12V fans exist (making assumptions about your solar panel).