I've received an email from the developer of the Rawkee plugin with this very useful information on how to properly export a Maya Binary to an X3D.

Under File, click the box next to "Export All".

1) Make sure the "Consolidate Files Locally" box is checked. - If this is
not checked, the exporter will use the path on your machine in the URL
field of the ImageTexture node. This option was put in because one of the
groups helping me debug the software wanted this.

2) Make sure the "Overwrite Existing Texture and Audio Files" is checked.
If you do not do this, the exporter will not write the image files to the
disk.

Warning, if "Consolidate Files Locally" box is NOT checked, but the
"Overwrite Existing Texture..." option is checked, the exporter will
overwrite the texture files that are referenced in the actual Maya texture
nodes... this will corrupt these files.

3) If the "Export Relative URLs" box is checked, then the exporter will
place the name of the texture in the "0" array location of the URL field of
the ImageTexture node. This is always the case.

4) If the "Export Relative URLs with Directory" box is checked, then the
exporter will place the relative path of the texture in the first open
array location of the URL field of the ImageTexture node.

Example: Consolidate Files Locally, Export Reltive URLs, and Export
Relative URLs with Directory are all checked

Exception: If "Consolidate Files Locally" is NOT checked, the relative path
will be ignored, and not be written to the X3D file.

5) If the "Export Base URLs" is checked, the exporter will append the
relative path of the texture to the full domain name (as entered in the
"URL Options" text box) and place this string in the first open array
location of the ImageTexture node's URL field.

Exception: If "Consolidate Files Locally" is NOT checked, the base url path
will be ignored, and not be written to the X3D file.

6) The "Export Textures" box must be checked, or no texture files will be
written to disk, and no ImageTexture nodes will be written to the X3D file.

7) The "Image File Sub-directory" text box contains the path to the all
"consolidated" texture files minus the actual texture file names. This path
is relative to the X3D file.

So, if you're using a Windows machine, and you save your X3D file to:
C:\testbed\testfile.x3d

And this text box contains the relative path "images\", then all the
texture files that the exporter writes to disk will be written to:
C:\testbed\images\

Within the X3D file, the "Relative URLs with Directory" info will be
written to the ImageTexture node's url field as:
"images/sometexture.jpg"

So, if you're using an OSX machine, and you save your X3D file to:
/Users/franco/Documents/testfile.x3d

And this text box contains the relative path "/images/", then all the
texture files that the exporter writes to disk will be written to:
/Users/franco/Documents//images/

Within the X3D file ImageTexture node's url filed, "Relative URLs with
Directory" will be written as:
"/images/sometexture.jpg"

8)Helpful Suggestion: Do not put a leading slash in the "Image File
Sub-directory" text box. If you're worried about exporting the full Base
URL path, put the a trailing slash in the "URL Options" text box.