If I remember my testing with KiCAD right it constructs ground planes by laying a grid on a board and filling in all the spots that are open or part of the ground net. (rounding corners as necessary ofc) The ground plane is thus made of many touching narrow traces and the DRC bot complains if the grid spacing (and thus trace width) is less than 8mill. Try a courser spacing ground plane and see if the DRC bot likes that better.
KiCad draws ground planes with a trace around the perimeter and then goes back and fourth with overlapping traces to fill the interior. Under ‘Zone Fill Options’, ‘Zone Min. thickness value’ controls the size of this trace. The defult may have been smaller than BatchPCB’s min.