Here's the answer I came up with.  In my HTML file, I removed the script that loads my *.nocache.js and instead generated it dynamically, like so:
<script type="text/javascript">
    function loadScript(scriptSrc)
    {
        var scriptTag = document.createElement('script');
        scriptTag.type = 'text/javascript';
        scriptTag.async = true;
        scriptTag.src = scriptSrc;
        var s = document.getElementsByTagName('script')[0];
        s.parentNode.insertBefore(scriptTag, s);
    }
    // Load the GWT script
    loadScript(('file:' == document.location.protocol ? "http://localhost:9876/" : "") + "admin/admin.nocache.js");
</script>
My Gradle task for running code Server looks like this:
task codeServer(dependsOn: "war") << {
    println("*----------------------------------------------------------------------------------------------*")
    println("   Ignore what this says below about going to http://localhost:9876/")
    println("   Instead, once the server below is up, in a separate command line, type:")
    println("       start $buildDir\\exploded\\Admin.html")
    println("*----------------------------------------------------------------------------------------------*")
    def gwtTempDir = "$buildDir/gwtTemp"
    (new File(gwtTempDir)).mkdirs()
    ant.java(classname: "com.google.gwt.dev.codeserver.CodeServer", failonerror: "true", fork: "true") {
        classpath {
            pathElement(location: "src/main/java")
            pathElement(location: "src/main/resources")
            pathElement(location: "$buildDir/classes/main")
            pathElement(path: configurations.compile.asPath)
        }
        jvmarg(value: "-Xmx512m")
        sysproperty(key: "java.util.logging.SimpleFormatter.format", value: System.getProperty("java.util.logging.SimpleFormatter.format"));
        arg(line: "-workDir " + gwtTempDir)
        arg(line: "-src " + "src/main/java")
        arg(value: "com.onlyinsight.oventio.Admin")
    }
}
So now, instead of having a second webserver, I can point my  browser to file:///F:/projects/ConferenceModule/build/exploded/Admin.html and it all works.  
I hope that helps somebody else.