In X3D, the "solid" field means exactly "do backface culling". So it's a straightforward convertion. See X3D specification https://www.web3d.org/documents/specifications/19775-1/V3.3/Part01/components/rendering.html#CommonGeometryFields
This was working in Blender 2.7 (the "backface culling" checkbox was then part of the "game settings").
It was broken at Blender 2.8 release.
Fixing it also makes X3D exporter consistent with glTF exporter, which also honors the "backface culling".
You can check resulting X3D files e.g. in my view3dscene ( http://michalis.ii.uni.wroc.pl/view3dscene-snapshots/ ) which supports "solid" field.