How to register an AppleScript as a potential default web browser?

Have you tried adding the CFBundleURLTypes and the sub-key LSIsAppleDefaultForScheme?

<key>CFBundleURLTypes</key>
<array>
    <dict>
        <key>CFBundleURLName</key>
        <string>Web site URL</string>
        <key>CFBundleURLSchemes</key>
        <array>
            <string>http</string>
            <string>https</string>
        </array>
        <key>LSIsAppleDefaultForScheme</key>
        <true/>
    </dict>
</array>

Not sure that it will help, but I have noticed these keys in the info.plist of all the apps that I have checked that show up in that browser list.