You set the normals of the "invisible" faces to the inside.
Blender usually ignores the normal when rendering, godot not.
Flip the normals of the faces where the normals currently show to the inner side. Then export again.
You get the mesh menu via "w". You'll also see that I enabled the normal display in the "Display" section.
B.t.w. (not needed here)
Double sided materials can be done by duplicating the faces and then flipping the normal. Note that blender may not display it properly if you assign different materials to both sides of a plane. (But godot does.) (there are also material flags in godot to let materials be double sided)