OK. I think the key thing that I was missing was that the website directory needed to be specified as an IIS application directory.
This was needed after modifying the web service's web.config file (under the systemServiceModel section) with:
<serviceHostingEnvironment multipleSiteBindingsEnabled ="true">
We moved the web service to run on the default port 80 just to potentially make the configuration easier. And it was the addition of the IIS headers for the port 80 that triggered the sequence of errors we had to fix. A header was needed for the internal IP and for the external -- well maybe we could have avoided all the trouble by just adding the external header -- except that such a change seemed to cause problems in earlier testing.
There was a side-effect of creating the main application directory. There were two http-modules we had to delete (rather than try to fix anything):
<remove name="RadUploadModule" />
<add name="RadUploadModule" type="Telerik.Web.UI.RadUploadHttpModule" preCondition="integratedMode" />
<remove name="RadCompression" />
<add name="RadCompression" type="Telerik.Web.UI.RadCompression" preCondition="integratedMode" />
We weren't using these, as best as I could tell, so I just removed these when the error popped up. It is interesting to see how the aspx website's web.config interjected itself in the process of calling the web service. But such error just reminds us that there is a hierarchy among the web directories.
Thanks for your interest and help.